https://karnatakaeducation.org.in/KOER/en/api.php?action=feedcontributions&user=Naveenkumar&feedformat=atomKarnataka Open Educational Resources - User contributions [en]2024-03-28T16:25:28ZUser contributionsMediaWiki 1.35.6https://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14845Circles Tangents Problems2014-08-14T05:29:06Z<p>Naveenkumar: /* Algorithm */</p>
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<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
Let ∠QBP=x˚<br><br />
<br />
∴∠QPB=x˚ (∵PQ=BQ)<br><br />
Now Let ∠PAQ=y˚<br><br />
∠QPA=y˚ (∵ PQ=AQ)<br><br />
∴In △PAB<br><br />
<br />
∠PAB+∠PBA+∠APB=180˚<br><br />
<br />
y˚+x˚+(x˚+y˚)=180˚<br><br />
2x˚+2y˚=180˚<br><br />
2(x˚+y˚)=180˚<br><br />
x˚+y˚=90˚<br><br />
∴ ∠APB=90˚<br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=<br />
A circle is touching the side BC of △ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ =<math>\frac{1}{2}</math> [perimeter of △ABC].<br />
[[File:123.png|300px]]<br />
<br />
==Algorithm==<br />
In the figure AQ , AR and BC are tangents to the circle with center O.<br><br />
BP=BQ and PC=CR (Tangents drawn from external point are equal) ---------- (1)<br><br />
<br />
Perimeter of △ABC=AB+BC+CA<br><br />
=AB+(BP+PC)+CA<br><br />
=AB+BQ+CR+CA ------ (From eq-1)<br><br />
=(AB+BQ)+(CR+CA)<br><br />
=AQ+AR ----- (From fig)<br><br />
=AQ+AQ -- --- (∵AQ=AR)<br><br />
=2AQ<br><br />
<br />
∴ AQ = <math>\frac{1}{2}</math> [perimeter of △ABC]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14842Circles Tangents Problems2014-08-14T05:28:22Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
<ggb_applet width="800" height="600" version="4.0" 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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
Let ∠QBP=x˚<br><br />
<br />
∴∠QPB=x˚ (∵PQ=BQ)<br><br />
Now Let ∠PAQ=y˚<br><br />
∠QPA=y˚ (∵ PQ=AQ)<br><br />
∴In △PAB<br><br />
<br />
∠PAB+∠PBA+∠APB=180˚<br><br />
<br />
y˚+x˚+(x˚+y˚)=180˚<br />
2x˚+2y˚=180˚<br />
2(x˚+y˚)=180˚<br />
x˚+y˚=90˚<br />
∴ ∠APB=90˚<br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=<br />
A circle is touching the side BC of △ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ =<math>\frac{1}{2}</math> [perimeter of △ABC].<br />
[[File:123.png|300px]]<br />
<br />
==Algorithm==<br />
In the figure AQ , AR and BC are tangents to the circle with center O.<br><br />
BP=BQ and PC=CR (Tangents drawn from external point are equal) ---------- (1)<br><br />
<br />
Perimeter of △ABC=AB+BC+CA<br><br />
=AB+(BP+PC)+CA<br><br />
=AB+BQ+CR+CA ------ (From eq-1)<br><br />
=(AB+BQ)+(CR+CA)<br><br />
=AQ+AR ----- (From fig)<br><br />
=AQ+AQ -- --- (∵∵∵∵∵∵∵∵∵∵∵∵∵∵∵∵∵∵AQ=AR)<br><br />
=2AQ<br><br />
<br />
∴ AQ = <math>\frac{1}{2}</math> [perimeter of △ABC]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14838Circles Tangents Problems2014-08-14T05:24:39Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
Let ∠QBP=x˚<br><br />
<br />
∴∠QPB=x˚ (∵PQ=BQ)<br><br />
Now Let ∠PAQ=Y˚<br><br />
∠QPA=Y˚ (∵ PQ=AQ)<br><br />
∴In △PAB<br><br />
<br />
∠PAB+∠PBA+∠APB=180˚<br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=<br />
A circle is touching the side BC of △ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ =<math>\frac{1}{2}</math> [perimeter of △ABC].<br />
[[File:123.png|300px]]<br />
<br />
==Algorithm==<br />
In the figure AQ , AR and BC are tangents to the circle with center O.<br><br />
BP=BQ and PC=CR (Tangents drawn from external point are equal) ---------- (1)<br><br />
<br />
Perimeter of △ABC=AB+BC+CA<br><br />
=AB+(BP+PC)+CA<br><br />
=AB+BQ+CR+CA ------ (From eq-1)<br><br />
=(AB+BQ)+(CR+CA)<br><br />
=AQ+AR ----- (From fig)<br><br />
=AQ+AQ -- --- (∵∵∵∵∵∵∵∵∵∵∵∵∵∵∵∵∵∵AQ=AR)<br><br />
=2AQ<br><br />
<br />
∴ AQ = <math>\frac{1}{2}</math> [perimeter of △ABC]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14828Circles Tangents Problems2014-08-14T05:20:20Z<p>Naveenkumar: /* == */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
Let ∠QBP=x˚<br><br />
<br />
∴∠QPB=x˚ (∵PQ=BQ)<br><br />
Now Let ∠PAQ=x˚<br><br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=<br />
A circle is touching the side BC of △ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ =<math>\frac{1}{2}</math> [perimeter of △ABC].<br />
[[File:123.png|300px]]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14827Circles Tangents Problems2014-08-14T05:19:57Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
Let ∠QBP=x˚<br><br />
<br />
====<br />
∴∠QPB=x˚ (∵PQ=BQ)<br><br />
Now Let ∠PAQ=x˚<br><br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=<br />
A circle is touching the side BC of △ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ =<math>\frac{1}{2}</math> [perimeter of △ABC].<br />
[[File:123.png|300px]]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14826Circles Tangents Problems2014-08-14T05:19:23Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
Let ∠QBP=x˚<br><br />
====<br />
∴∠QPB=x˚ (∵PQ=BQ)<br><br />
Now Let ∠PAQ=x˚<br><br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=<br />
A circle is touching the side BC of △ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ =<math>\frac{1}{2}</math> [perimeter of △ABC].<br />
[[File:123.png|300px]]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14825Circles Tangents Problems2014-08-14T05:18:24Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
Let ∠QBP=x˚<br><br />
∴∠QPB=x˚ _ (∵PQ=BQ)<br><br />
Now Let ∠PAQ=x˚<br><br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=<br />
A circle is touching the side BC of △ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ =<math>\frac{1}{2}</math> [perimeter of △ABC].<br />
[[File:123.png|300px]]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14824Circles Tangents Problems2014-08-14T05:18:08Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
Let ∠QBP=x˚<br><br />
∴∠QPB=x˚ (∵PQ=BQ)<br><br />
Now Let ∠PAQ=x˚<br><br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=<br />
A circle is touching the side BC of △ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ =<math>\frac{1}{2}</math> [perimeter of △ABC].<br />
[[File:123.png|300px]]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14823Circles Tangents Problems2014-08-14T05:17:19Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
Let ∠QBP=x˚<br><br />
∴∠QPB=x˚ (∵PQ=BQ)<br><br />
Now Let ∠PAQ=x˚<br><br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=<br />
A circle is touching the side BC of △ABC at P. AB and AC when produced are touching the circle at Q and R respectively. Prove that AQ =<math>\frac{1}{2}</math> [perimeter of △ABC].<br />
[[File:123.png|300px]]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14807Circles Tangents Problems2014-08-14T04:51:16Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br><br />
b)<br><br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14806Circles Tangents Problems2014-08-14T04:50:27Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
a)<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==<br />
=Problem 5=</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14804Circles Tangents Problems2014-08-14T04:49:20Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
'''Step-1''''''Bold text'''<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.<br />
<br />
=problem 3 [Ex-15.2 B.7]=<br />
Circles <math>C_{1}</math> and <math>C_{2}</math> touch internally at a point A and AB is a chord of the circle<math>C_{1}</math> intersecting <math>C_{2}</math> at P, Prove that AP= PB.<br><br />
[[Image:problem 3 on circle.png|300px]]<br />
==Concepts used==<br />
#The radii of a circle are equal<br />
#Properties of isosceles triangle.<br />
#SAS postulate<br />
#Properties of congruent triangles.<br />
<br />
==Prerequisite knowledge==<br />
#The radii of a circle are equal.<br />
# In an isosceles triangle angles opposite to equal sides are equal.<br />
#All the elements of congruent triangles are equal.<br />
<br />
==Algoritham==<br />
In ∆AOB <br><br />
AO=BO [Radii of a same circle]<br><br />
∴ ∠OAB = ∠OBA --------------I [∆AOB is an isosceles ∆}<br><br />
Then,<br> <br />
In ∆AOP and ∆BOP,<br><br />
AO = BO [Radii of a same circle]<br><br />
OP=OP [common side]<br><br />
∠OAP = ∠OBP [ from I]<br><br />
∴ ∆AOP ≅ ∆BOP [SAS postulate] <br><br />
∴ AP = BP [corresponding sides of congruent triangles ]<br />
<br />
=problem-4=<br />
'''In the given Quadrilateral ABCD , BC=38cm , QB=27cm , DC=25cm and AD⊥DC find the radius of the circle.(Ex:15.2. A-6)'''<br><br />
[[File:fig3.png|200px]]<br />
==Interpretation of problem==</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14420Circles Tangents Problems2014-08-12T11:43:20Z<p>Naveenkumar: /* Concepts used */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br><br />
[[https://www.geogebratube.org/student/m86017 '''Click here for geogebra animation''']]<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
'''Step-1''''''Bold text'''<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14413Circles Tangents Problems2014-08-12T11:37:24Z<p>Naveenkumar: /* Concepts used */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
#Axiom-1:- "Things which are equal to same thing are equal"<br />
<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
'''Step-1''''''Bold text'''<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14412Circles Tangents Problems2014-08-12T11:35:22Z<p>Naveenkumar: /* Algorithm */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
==Algorithm==<br />
In the above figure AB is direct common tangent to two circles and PQ is the Transverse common tangent.<br><br />
'''Step-1''''''Bold text'''<br><br />
AQ=QP and BQ=QP (Tangents drawn from external point are equal)<br><br />
By axiom-1, AQ=BQ<br><br />
∴tangent at P bisects AB at Q.</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14410Circles Tangents Problems2014-08-12T11:29:23Z<p>Naveenkumar: /* Concepts used */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#Angle subtended by equal sides are equal.<br />
==Algorithm==</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14408Circles Tangents Problems2014-08-12T11:23:22Z<p>Naveenkumar: /* Concepts used */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle <br> a) are equal <br> b] subtend equal angle at the center <br> c] are equally inclined to the line joining the center and external point.<br />
#</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14396Circles2014-08-12T11:04:19Z<p>Naveenkumar: /* Hints for difficult problems */</p>
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<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
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<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
<ggb_applet width="1280" height="600" version="4.0" 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*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠OPQ <br><br />
[[http://karnatakaeducation.org.in/KOER/en/index.php/Class10_circles_tangents_problems '''Solution''']]<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
angle<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-angle<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14367Circles Tangents Problems2014-08-12T10:36:28Z<p>Naveenkumar: /* Concepts used */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
<ggb_applet width="800" height="600" version="4.0" 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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#The tangent drawn from an external point to a circle are equal<br />
#</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14337Circles Tangents Problems2014-08-12T10:11:55Z<p>Naveenkumar: /* Interpretation of the problem */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these circles.<br />
#PQ is the transverse common tangent drawn to these circles at point P.<br />
#Using the tangent properties students have to show AQ=BQ and ∠APB=90°<br />
==Concepts used==<br />
#</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14320Circles Tangents Problems2014-08-12T10:04:00Z<p>Naveenkumar: /* Problem-2 */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]<br />
==Interpretation of the problem==<br />
#In the given figure two circles touch externally.<br />
#AB is the direct common tangent to these</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14307Circles Tangents Problems2014-08-12T09:54:40Z<p>Naveenkumar: /* Problem-2 */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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PNUqEtLxWO+vmPnaRrWQqZn8stI0wNpNGKekiorxf0aGTWkqbebfXcZHnUChM6XDpwfz6s+9BrjvWc0rzv77WZ63kf7LLtvWuE+DTg8KztE87WZ3nyrA2IqmlKo7UY+Go8lsyBClgR46ryijPtCagpOQ7uiEME/HmOOvvEsU4WFk2iH1+mAxETj/q5dQKbwJ0gjCqm2iI+nuBXA29v4Pi8v8ZWAoXOUsk0mqKPJHQiKlgHzKtVKEnCkkycHw0RAyrXISd5uuR71gmajQ5F9dxleddN98qEQfOQorOse/p6xvO7st4/geR/tuqN91nDfSowg0baZ3vxpBYhcSTFT6enplr94dLSJ9UFvrEEk/ST88aGhYpNq3z6xtqGaTVrSoLhYBIhSRsbgfOl9IQPj4oOCBucrAUIA3bsnNsG6ZKylaOrtlutRJ5njVTe9hiRySF55PPL3GMvrzn77CE2h0bz5432WoCB47jWIza8u8yImbAoQqy+vTIU+IQEgMJVKsQkVGGg8f+2a+HR/IDRRdu8Wa4mwMKFqPzR+Yc7KElfNaWoy9kni4oxQuLgAzQNtdlq6bcMGoLt7/O8sebu9suR71KWnvVyvuqlHXapBKIqXmjqq7uHvkbzu1MyKqx3sdZf6FXI10mfjBmomAsNF+K+ahdfOPuGepQj3zA4BoTlsqSkVHBxsmzFT69cDRUW2acaZ4V71aQ2I6Wg+PgEbFwfE0F/V4uDBgzxmiosDMlTUjJJjmeLimlGAJAidX+qAZ2Zm8n+FiwMiSbJM3eDrHnNxQIyqrKy0bcsUFwdkKkSWKRrN5+/vzy1TXBwQSZJlitYN12g0/M5zcUBI1HyiZhTNGsItU9ZTzSZ/qHYcgcrrGFouxbJjuscq1PueNrxHc/Iq2m6mQOUZiP72sT3//S3taD5/E+3x90c8b8hLuKaU19DvYM717BIQhULBOuB80crx1dAw+SH6fepm9OSLD6Uap++hq1CiNfI21AcusmPq/edQ+eVutt0edw+a41fYdvPFGLSEJYyZd1tUMjrvPEDp4jXDrkd5NQSIMWVNZ2+gJTxx2Heo+trXrOvZFSCSZUoxdAwDFxwcHFjq6ekxHNPr9WxYsH4getf0HL13Inqq60VHojEY8uH7G9h+07mbeNrbh95KFTqSsvBw2TrRaLJ6F3pr1Gy7bs8ptCvSZAFInxt6PTqmLRYDFeu8g9CRcH/Yd6j8wsvs69k8INSE4papscGQ0qxZsxgQfX19bH/NGvFJ3NbWZjg3e/Zsds7R0ZFtm6O26NRB+2X//dWg/cYjoWi9moTiOavY/qOV37JCrS2tYgVZ3y0aULruF6LpTJQh6R4ZByBpTkRAW1Qx7HqUFzXB2PYnrgyeod9htOvZJSBkmaLON3XCuWXK+qImDT21u9Lz0adsYO380oVfGgoqFebmkFg8aWhG4WsroSuvYceoCVTxmdugwt6ZkovGwHBD0paJiwxR3vn/+BEqfv/t8OsJn2/48SJLXffEkPaRvsNI17MrQEwtU3xpAPsQdaBrtxyw2+tZBBDJMsVjprjsvmls7gcoJMTJyYlbprg4IKaiIEKyTFFQIRcXB2RAppYpLi4OyIAoHIRbprg4IENEMFAgIQUUcssUFwdkQKaWKQpF5+LiAv4f68D+M77/+KgAAAAASUVORK5CYIJQSwcIO9ZP2FcTAABSEwAAUEsDBBQACAAIAH2G6UQAAAAAAAAAAAAAAAAWAAAAZ2VvZ2VicmFfamF2YXNjcmlwdC5qc0srzUsuyczPU0hPT/LP88zLLNHQVKiuBQBQSwcI1je9uRkAAAAXAAAAUEsDBBQACAAIAH2G6UQAAAAAAAAAAAAAAAAMAAAAZ2VvZ2VicmEueG1s7Vvdctu4Fb7efQoMZ6fTbSMKIAiQTOXseDO708xk187a7ex0Zy8oEpIQU6SWpGzZsw/QN+htX6Gb5Al6n4fok/QAIClSkn8k24nSqSc2SOIAB/i+cw4OQGbw1WKaoHORFzJLDyxiYwuJNMpimY4PrHk56vnWV88+H4xFNhbDPESjLJ+G5YHlKkkZH1geiZjvh7g3pJz3XM5YLxz5w17kYcJ8l/BgxCyEFoV8mmbfh1NRzMJInEQTMQ1fZlFYasWTspw97fcvLi7sWpWd5eP+eDy0F0VsIRhmWhxY1cVT6K7T6IJqcQdj0v/xu5em+55MizJMI2EhNYW5fPb5Z4MLmcbZBbqQcTmBCVPuW2gi5HgCk/IcYqG+kpoBIjMRlfJcFNC2dasnXU5nlhYLU1X/mblCSTMfC8XyXMYiP7CwTdyAOI6PARGPudQHPLJcirSshGul/bq7wbkUF6ZfdaVVuhYqsywZhqpL9OuvyMEORk9UQUzhQMG5qcLmGaamcEzhmoIZGdc0d42oa2RcI+NSC53LQg4TcWCNwqQADGU6yoG/5r4oLxOhx1M9WE6fPIE5FfIKhCkGQzGgw3OMn6hfDr+uquh3J0laWst8vqXSWiUhzLm7Tuc+OmmtEyBbV+mwa6bJb0DXjOFO82QtaEGV/qd/1zRSZwuN5v5+Crn7QaY46NeuMqi8AxUTJVsxWYppofyFBogFyuwJYuAb3AMrZ4gEUHgOAm9AhCGXwS3xEVelh6gHFS6iyEdKjlCknYP58Mf1dGccMehMPfXAJxEBRS5iFBHtUy4CT0LaL8FHHQoSjCEGjZR64qguKEcuhzvqIxfGqFzSIyBIoSHcg3oHUYKoakw85HDEVX/EVa7OfTV06NJBHCNOVIfg1eDRxptB3kdUzYZXcMl0Ni87EEXTuL4ss1nDBUhDPFqGPROfOlHxs0ESDkUCK8WJYhKh8zBRHqEVjbK0RDWJjnk2zsPZREbFiShLaFWg1+F5+DIsxeJbkC5q3Vo2ytLiOM/K51kyn6YFQlGW4GbMWUJa104zarihrQq3XcFaFbx17W3Um0ENmhcC9Gd5UYuHcfxCSSxDAyB5lCaXX+ciPJtlsjuNQV8vOgMxjxIZyzD9Kxir0qJwQc0apMNVvQYx7tUjyfL45LIAE0aLv4k8gyBDXDto/3gWujRVlDAbt3+A8SIKlfe5QbeRD1WXVR3D3VbE6BbnDUfhQiynO86Vb7duXhRfZ8nykUbgeTgr57lOICA85mpah+k4EdpKtG/D6hydDbPFiTEPavo6vZzBHTYjGI418giig8NgwRxX5dCUWkYNrZHCWgZrCVzbm4ybehI4WkKXQ1NqKTBgM7RqqqSeJsG1GlnomIatynPqeKXMX63181SWL+ubUkZny6mqBt/Pp0PRGFG3T/JQfQ76K1Y2OBN5KpLKqIHMeTYvjI+27D0WkZzCramoIAkVXX+BAZinsRjnoh54opMzA5iuxW1zXXusu/o2z6Yv0vNTsIWVAQz69SgHRZTLmbI5NISF4EwsrSqWRQjrSNxup7wQph6p9QLgKRU04J/zcpLlOv2CsAKlcr5ETCHXQqU2r3Q+FbmMGqBDncfBoObVuLkdmJErlFE2fA3xrlkSTZsljVB9jQGiMJlNQp39VWYWXoq8A43u7bssrhRXckWi0kY0lalOTabhQlsiCocFhMISMmfgIl1mzmZkdSjBWOXl0MTzdYZ+qd0fLkZyIZpADxjJK7CJsDOZpSOUEKXPIBUt9BDKyi/1xZ9lHIu0GW2YgvVoDiBOzcx0EawRwph203QG09cRocV8RcytFA1XKaI23RuKSEUR25ohH9cM+Z8aQzrMN1gfWfflQkf2hg18Rzbw5glumB7ZNL0qZBWKC2y7rqaC2AT2g1dmJ2x2gmquaqHqpDfm6Uq0awMWZdNpmMYo1QnvsQZsmYCFkMo8l3mUiJ+OnqDw5wqeeVlXHpr+ql5uIeBwJwKIY5ZVXe4DCZ5N3W4u8oiMnIixer7CyREgQCr0O3xEN/NRVL3ViEe3MNKa/13i05ZktO3aRBi1qCkMe75NXGc939slsIhfUtOkMImJnM4SGcmywS5RXL9IS0hThF6n17OPMyFmKu07Sk/zMC3U4ZKRaWU11xBo3Gczf8M1/uKb+YNsorXgxLuxp48AGv4IdnZYYfANfLBb+JBjkZ7DaGG/gtACV6d5l7gyg6v6yQIQ6hnLINWjK9LyL3DkXC7QYS1/WEsdwkarF9gU0/ZuAp7TSsWhq3tWsfQQtls97bsbjUVtI+VIRjdzfArZ47qTHhqS4zWShakY3cy1wrahUqxT3U3EP5Sn9hzbD3xCfB/7AC9jVDtuj9nUIR4NCMUudSGr037seDZjxHddx/EYZx6nj+PH16cAHRRH+4LiViD29G7aYTTwvQBquec+PopdA2+i44qJx8aSlUE7jYqWpR9vkx4c75QecBPMVDE0xUPQg4nvcOzzAKwWKDEbFBt7bdJUPvLg6/0tSI+uQ/rVNki/+rBIH41GhSj1Oq+BpOQ+NEC09kiHh2CPePhmGx6+2XOL/zBQ/xBebl45X62hO74Z3Rx6qrEb70lq+4ih/rZE6+MmvtfSerxG6+TutE72hNZHzIP2m9YbN6Tr1MrtNqRyv+klNvNWwuSV3kow3yMeYwy2HOo1yafrtTfSux6QX29H7+s9oXcrdqlNqMccTDCBao8Fnyy5+vXWdZ6rE5r1E6X3/7qZYv3ipCEQpFX71iG4AhV7AacO9wLXDzg3KnY7hyV491PxtfdoWxxO51Ery6nPEpMku/hBjBKx0MDeh4Um49EsHK2z8NtWLPz2fxZ2C3TH1wW6s+0C3dmeBDpmA+kO9TgB3gOi3rZf1udsKt0kNg98SgLmOdz3mH8TO596bFsn9f2brbzqzapXYdsNfO5RSPQwJ36AqffQbuWwO26te66vuQ0e1ev4I8S+49ti39utWHr7EVgid2aJVUcgW7Pk7gdLh9f60rutWHq3zlIAO16X88AlELB8/PAkPcAKtRULbFsWxGKWgxr1erKeo1iUxEJQcWD97pd5Vv7pC/Sfv//z6PjVgblFf0S/fxmeih9/ev/m5y/hrnr8hSnNB3JdJ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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br><br />
[[File:fig2.png|400px]]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14305Circles Tangents Problems2014-08-12T09:54:06Z<p>Naveenkumar: /* Problem-2 */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)<br />
[[File:fig2.png|400px]]</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=File:Fig2.png&diff=14304File:Fig2.png2014-08-12T09:53:55Z<p>Naveenkumar: MsUpload</p>
<hr />
<div>MsUpload</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14293Circles Tangents Problems2014-08-12T09:44:41Z<p>Naveenkumar: /* Problem-2 */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=<br />
In the figure two circles touch each other externally at P. AB is a direct common tangent to these circles. Prove that<br>a). Tangent at P bisects AB at Q<br> b). ∠APB=90° (Exescise-15.2, B.3)</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14286Circles Tangents Problems2014-08-12T09:38:53Z<p>Naveenkumar: </p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ<br />
=Problem-2=</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14284Circles Tangents Problems2014-08-12T09:37:17Z<p>Naveenkumar: /* Interpretation of the problem */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
<ggb_applet width="800" height="600" version="4.0" 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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14282Circles Tangents Problems2014-08-12T09:36:26Z<p>Naveenkumar: /* Interpretation of the problem */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A. #OP and OQ are the radii. #The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
<ggb_applet width="800" height="600" version="4.0" 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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles_Tangents_Problems&diff=14279Circles Tangents Problems2014-08-12T09:35:18Z<p>Naveenkumar: /* Problem 1 */</p>
<hr />
<div>=Problem 1=<br />
Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:image_circle_with_tangents.png|300px]]<br />
==Interpretation of the problem==<br />
O is the centre of the circle and tangents AP and AQ are drawn from an external point A. OP and OQ are the radii. The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Geogebra file===<br />
<ggb_applet width="800" height="600" version="4.0" 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<br />
==Concepts used==<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
==Algorithm==<br />
OP=OQ ---- radii of the same circle<br />
OA is joined.<br><br />
In quadrillateral APOQ ,<br><br />
∠APO=∠AQO=<math>90^{0}</math> [radius drawn at the point of contact is perpendicular to the tangent]<br><br />
∠PAQ+∠POQ=<math>180^{0}</math><br> <br />
Or, ∠PAQ+∠POQ=<math>180^{0}</math><br><br />
∠PAQ = <math>180^{0}</math>-∠POQ ----------1<br><br />
Triangle POQ is isoscles. Therefore ∠OPQ=∠OQP<br><br />
∠POQ+∠OPQ+∠OQP=<math>180^{0}</math><br><br />
Or ∠POQ+2∠OPQ=<math>180^{0}</math><br><br />
2∠OPQ=<math>180^{0}</math>- ∠POQ ------2<br><br />
From 1 and 2 <br><br />
∠PAQ=2∠OPQ</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14275Circles2014-08-12T09:34:11Z<p>Naveenkumar: /* Problem-1 */</p>
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<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
<ggb_applet width="1280" height="600" version="4.0" 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<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
<ggb_applet width="1280" height="600" version="4.0" 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*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠OPQ <br><br />
[[http://karnatakaeducation.org.in/KOER/en/index.php/Class10_circles_tangents_problems '''Solution''']]<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14269Circles2014-08-12T09:27:30Z<p>Naveenkumar: /* Problem-1 */</p>
<hr />
<div><!-- This portal was created using subst:box portal skeleton --><br />
<!-- BANNER ACROSS TOP OF PAGE --><br />
{| id="mp-topbanner" style="width:100%;font-size:100%;border-collapse:separate;border-spacing:20px;"<br />
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|style="width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]<br />
|}<br />
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].<br />
<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
<ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
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kNFzN5dGgAAABgAAAAWAAAAAAAAAAAAAAAAAPQDAABnZW9nZWJyYV9qYXZhc2NyaXB0LmpzUEsBAhQAFAAIAAgA8q5mQxAhsry/CQAA0CoAAAwAAAAAAAAAAAAAAAAAUgQAAGdlb2dlYnJhLnhtbFBLBQYAAAAAAwADAMIAAABLDgAAAAA=" enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠OPQ <br><br />
[[File:fig1.png|200px]]<br />
===Interpretation of the problem===<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Concepts used.===<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
===Algorithm===<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14267Circles2014-08-12T09:26:34Z<p>Naveenkumar: /* Problem-1 */</p>
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<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
<ggb_applet width="1280" height="600" version="4.0" 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*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠OPQ <br><br />
[[File:fig1.png|200px]]<br />
===Interpretation of the problem===<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Concepts used.===<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14266Circles2014-08-12T09:26:03Z<p>Naveenkumar: /* Problem-1 */</p>
<hr />
<div><!-- This portal was created using subst:box portal skeleton --><br />
<!-- BANNER ACROSS TOP OF PAGE --><br />
{| id="mp-topbanner" style="width:100%;font-size:100%;border-collapse:separate;border-spacing:20px;"<br />
|-<br />
|style="width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]<br />
|}<br />
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].<br />
<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
<ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
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kNFzN5dGgAAABgAAAAWAAAAAAAAAAAAAAAAAPQDAABnZW9nZWJyYV9qYXZhc2NyaXB0LmpzUEsBAhQAFAAIAAgA8q5mQxAhsry/CQAA0CoAAAwAAAAAAAAAAAAAAAAAUgQAAGdlb2dlYnJhLnhtbFBLBQYAAAAAAwADAMIAAABLDgAAAAA=" enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:fig1.png|200px]]<br />
===Interpretation of the problem===<br />
#O is the centre of the circle and tangents AP and AQ are drawn from an external point A.<br />
#OP and OQ are the radii.<br />
#The students have to prove thne angle PAQ=twise the angle OPQ.<br />
===Concepts used.===<br />
#The radii of a circle are equal.<br />
#In any circle the radius drawn at the point of contact is perpendicular to the tangent.<br />
#The tangent drawn from an external point to a circle a] are equal b] subtend equal angle at the centre c] are equally inclined to the line joining the centre and extrnal point.<br />
#Properties of isoscles triangle.<br />
#Properties of quadrillateral ( sum of all angles) is 360 degrees<br />
#Sum of three angles of triangle is 180 degrees.<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14257Circles2014-08-12T09:21:00Z<p>Naveenkumar: /* Hints for difficult problems */</p>
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<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
<ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
<ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:fig1.png|200px]]<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=File:Fig1.png&diff=14255File:Fig1.png2014-08-12T09:20:10Z<p>Naveenkumar: MsUpload</p>
<hr />
<div>MsUpload</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14254Circles2014-08-12T09:19:29Z<p>Naveenkumar: /* Problem-1 */</p>
<hr />
<div><!-- This portal was created using subst:box portal skeleton --><br />
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[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]<br />
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[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]<br />
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[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]<br />
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[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]<br />
|}<br />
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].<br />
<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
<ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
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kNFzN5dGgAAABgAAAAWAAAAAAAAAAAAAAAAAPQDAABnZW9nZWJyYV9qYXZhc2NyaXB0LmpzUEsBAhQAFAAIAAgA8q5mQxAhsry/CQAA0CoAAAwAAAAAAAAAAAAAAAAAUgQAAGdlb2dlYnJhLnhtbFBLBQYAAAAAAwADAMIAAABLDgAAAAA=" enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:fig1.png|400px]]<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14251Circles2014-08-12T09:16:37Z<p>Naveenkumar: /* Problem-1 */</p>
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<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
<ggb_applet width="1280" height="600" version="4.0" 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*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ <br><br />
[[File:Screenshot from 2014-08-12 14:43:11.png|100px]]<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14250Circles2014-08-12T09:15:58Z<p>Naveenkumar: /* Problem-1 */</p>
<hr />
<div><!-- This portal was created using subst:box portal skeleton --><br />
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[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]<br />
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[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]<br />
|}<br />
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].<br />
<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
<ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
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kNFzN5dGgAAABgAAAAWAAAAAAAAAAAAAAAAAPQDAABnZW9nZWJyYV9qYXZhc2NyaXB0LmpzUEsBAhQAFAAIAAgA8q5mQxAhsry/CQAA0CoAAAwAAAAAAAAAAAAAAAAAUgQAAGdlb2dlYnJhLnhtbFBLBQYAAAAAAwADAMIAAABLDgAAAAA=" enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ<br />
[[File:Screenshot from 2014-08-12 14:43:11.png|100px]]<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14249Circles2014-08-12T09:15:33Z<p>Naveenkumar: /* Problem-1 */</p>
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<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
<ggb_applet width="1280" height="600" version="4.0" 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*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ<br />
[[File:Screenshot from 2014-08-12 14:43:11.png|300px]]<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14248Circles2014-08-12T09:14:52Z<p>Naveenkumar: /* Problem-1 */</p>
<hr />
<div><!-- This portal was created using subst:box portal skeleton --><br />
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[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]<br />
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[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]<br />
|}<br />
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].<br />
<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
<ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
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kNFzN5dGgAAABgAAAAWAAAAAAAAAAAAAAAAAPQDAABnZW9nZWJyYV9qYXZhc2NyaXB0LmpzUEsBAhQAFAAIAAgA8q5mQxAhsry/CQAA0CoAAAwAAAAAAAAAAAAAAAAAUgQAAGdlb2dlYnJhLnhtbFBLBQYAAAAAAwADAMIAAABLDgAAAAA=" enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ<br />
[[File:Screenshot from 2014-08-12 14:43:11.png|400px]]<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=File:Screenshot_from_2014-08-12_14-43-11.png&diff=14247File:Screenshot from 2014-08-12 14-43-11.png2014-08-12T09:14:39Z<p>Naveenkumar: MsUpload</p>
<hr />
<div>MsUpload</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Circles&diff=14245Circles2014-08-12T09:09:02Z<p>Naveenkumar: /* Hints for difficult problems */</p>
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<br />
= Concept Map =<br />
<mm>[[circle.mm|flash]]</mm><br />
<br />
= Textbook =<br />
==ncert books==<br />
[http://ncert.nic.in/NCERTS/textbook/textbook.htm?iemh1=10-15 Class 9 Mathematics] contain simple description and theorems on circles.<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
#[http://www.mathsisfun.com/geometry/circle.html maths is fun] A good website on definitions for circles.<br />
#[http://www.coolmath.com/reference/circles-geometry.html Cool math] For clear and easy definitions.<br />
#[http://www.mathopenref.com/circle.html Open reference] Contains good simulations.<br />
#[http://en.wikipedia.org/wiki/Math_circle#History Wikipedia] Has good explanations on circles.<br />
#[http://www.khanacademy.org Khan academy] Has good educative videos.<br />
#[http://www.arvindguptatoys.com Arvind gupta toys] Contains good information. <br />
#[http://nrich.maths.org/2490 nrich.maths.org] Refer for understanding Pi.<br />
<br />
==Reference Books==<br />
#[http://archive.org/stream/schoolgeometry00hall#page/n11/mode/2up School Geometry] By Hall and Stevens. Part3 pageno 143. Contains basic definitions and proofs given by Euclid.<br />
<br />
= Teaching Outlines =<br />
Introduction to circle<br />
<br />
==Concept #1 Introduction to Circle==<br />
<br />
===Notes for teachers ===<br />
Source: http://circlesonly.wordpress.com/tag/inventions/<br><br />
Summary :<br />
The circle is the most primitive and rudimentary of all human inventions, and at the same time, the most dynamic. It is the cornerstone in the foundation of science and technology. It is the basic tool of all engineers and designers. It is used by the greatest artists and architects in the history of mankind. Without a circular shape the wheel, pulleys, gears, ball bearings and a thousand other items we take for granted wouldn’t exist. And of course we would never have the pleasure of driving a car, riding a giant wheel, or watching the moon landing on our television set.<br><br />
If you look through any old patent claim, you will most likely find the repeated use of circles, spheres, curves, arches, etc. circles are everything and they are nothing. They don’t exist in reality and yet they are the basis of all that mankind has brought into existence. That is why a circle is so fantastic.<br />
<br />
===Learning objectives===<br />
# Appreciation of circle as an important shape as it is an intrical component in the invention of almost everything that we see around us.<br />
# To make students know that circle is a 2-dimensional plane circular figure.<br />
# All points on its edge are equidistant from the center.<br />
# The method of drawing a circle<br />
# The size of the circle is defined by its radius.<br />
# To elicit the difference between a bangle or a circular ring and circle as such.<br />
<br />
<br />
[[File:circle.jpeg|200px]]<br />
<br />
===Activity No # 1. A discussion on “Life without circular shaped figures.”===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 45 minutes<br />
*Materials/ Resources needed: Paper, pen <br />
*Prerequisites/Instructions, if any:<br />
Previous day homework : <br />
# Ask the children to make a list of all circular objects that they can think of :<br />
# List as many devices as you can think of that depend on the wheel.(Consider objects in your home, at school, games and toys, machines, vehicles and engines as you make your list.)<br />
# Now imagine living in a world without any kind of wheels or rolling devices. How would life be different? Would it be harder? How and why? Describe what it would be like to live without any wheels.<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
<br />
*Process:(How to do the activity)<br />
Have an open discussion with children. Initially let the children share their ideas and do most of the talking. Ensure that the intended discussion remains within the context. Make a mind map on blackboard of all relavant points discussed . Let them appreciate the significance of circular shape thus setting the stage for further study of this fantastic shape called “circles”.<br />
*Developmental Questions :(What discussion questions)<br />
# What all shapes do we see around us ?<br />
# Can you imagine bicycles and your other vehicles without circular wheels ?<br />
# How different life would have been if wheel was not disovered ?<br />
# What about potter's wheel and stone mill?<br />
# Do you think that it is necessary for us to study and understand the parameters of circle in depth and detail ?<br />
*Evaluation:<br />
# Do you all now agree that wheel is one of the greatest inventions of mankind? Justify<br />
*Question Corner:<br />
# Are shapes important ? How?<br />
# Is bangle a circle ? <br />
# When you say shape, what do you mean ?<br />
<br />
===Activity No # 2. ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time<br />
*Materials/ Resources needed<br />
*Prerequisites/Instructions, if any<br />
*Multimedia resources<br />
*Website interactives/ links/ / Geogebra Applets<br />
*Process/ Developmental Questions<br />
*Evaluation<br />
*Question Corner<br />
<br />
===Activity No # 3. Circle of varying radius using Geogebra ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 20 mins<br />
*Materials/ Resources needed: Laptop, geogebra,projector and a pointer<br />
*Prerequisites/Instructions, if any:<br />
# Theory on circles introduction should have been done. <br />
*Multimedia resources: Laptop<br />
*Website interactives/ links/ / Geogebra Applets:<br />
This geogebra file has been made by ITfC-Edu-Team.<br />
<ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
<br />
*Process:<br />
Show the geogebra file and ask the following questions.<br />
*Developmental Questions:<br />
# What is a circle ?<br />
# Which point is the centre of the circle ?<br />
# What is the radius of this circle ?<br />
# How do you name the radius ?<br />
*Evaluation:<br />
#By what parameter is the size of a circle defined ?<br />
#Bigger the radius, _____________ <br />
*Question Corner:<br />
# How do you name a circle ?<br />
# Can you draw a circle without knowing the radius ?<br />
<br />
===Activity No # 3. Is circle a Polygon ? - A debate.===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
''[http://karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div><br />
|}<br />
*Estimated Time: 40 minutes<br><br />
*Materials/ Resources needed<br />
Laptop, geogebra file, projector and a pointer.<br><br />
*Prerequisites/Instructions, if any<br />
# Ensure that lesson on polygons is done.<br />
*Multimedia resources:Laptop <br />
*Website interactives/ links/ / Geogebra Applets: This geogebra file has been created by maths STF teachers.<br />
<ggb_applet width="1280" height="600" version="4.0" 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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="false" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="false" /><br />
*Process:<br />
Demonstrate the geogebra file and ask the questions listed below.<br />
*Developmental Questions:<br />
# How many sides does this figure have ?<br />
# Name the figure formed.<br />
# What is hapenning to the length of the sides as the number of sides is increased ?<br />
# What shape is this ?<br />
# So, can circle be considered a polygon ? Justify<br />
*Evaluation:<br />
# Are the students able to comprehend that the number of sides is getting infinite as the shape resembles a circle ?<br />
# Are the students able to appreciate the application of polygon anology to circles.<br />
*Question Corner;<br />
Debate between two groups with these two perspectives.<br><br />
#Circle seems to have derived from polygons . Circle can be considered a polygon.<br />
Vs <br />
#A polygon is defined by a certain number of sides having non zero length. Then how can circle be a polygon ? (hint: all radii in a circle should be equal ???)<br />
<br />
= Hints for difficult problems =<br />
== Problem-1==<br />
#Tanjents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ<br />
<br />
== Ex 4.4.2==<br />
#Suppose two chords of a circle are equidistant from the centre of the circle, prove that the chords have equal length.<br />
'''DATA''' :- Let AB & CD are the two chords which are equidistant from the centre 'O' of the circle. [ Here OP is the perpendicular distance from the centre O to the chord AB and OQ is the perpendicular distance from the centre O to the chord CD] OP = OQ.<br />
<br />
'''TO PROVE :-''' AB = CD,<br />
<br />
'''CONSTRUCTION :-''' Join OA & OD.<br />
<br />
'''PROOF :-'''<br />
{[Consider In ∆AOP & ∆DOQ <br />
OA = OD<br />
OP = OQ<br />
Angle APO = Angle DQO<br />
∆AOP ≡ ∆DOQ <br />
AP = DQ<br />
Let AB = AP + BP<br />
= AP + AP<br />
= 2AP<br />
AB = 2DQ ---------- 1.<br />
and CD = CQ + DQ<br />
= DQ + DQ<br />
CD = 2DQ --------- 2.<br />
From equtn 1 & equtn 2<br />
AB = CD<br />
<br />
Radii of the circle<br />
Equi distances from circle<br />
<br />
SAS Axiom<br />
Acording to properties of SAS axiom.<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. AP = BP.<br />
<br />
<br />
Perpendicular drawn from centre to chord which <br />
bisect the chord, i.e. CQ = DQ<br />
Acording to AXIOM-1]}<br />
<br />
<br />
{|class="wikitable"<br />
|-<br />
|'''Steps'''<br />
|'''Explanation'''<br />
|-<br />
|[[Image:solution.png|300px]]<br />
|Explanation for thestep<br />
|-<br />
|Write the step<br />
|Explanation for thestep<br />
|}<br />
|}<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=File:Screenshot_from_2014-07-08_06_50_36.png&diff=14244File:Screenshot from 2014-07-08 06 50 36.png2014-08-12T09:07:40Z<p>Naveenkumar: MsUpload</p>
<hr />
<div>MsUpload</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Operations_on_sets&diff=11851Operations on sets2014-07-10T08:18:49Z<p>Naveenkumar: /* Assessment activities for CCE */</p>
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]<br />
|}<br />
While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].<br />
<br />
= Concept Map =<br />
<mm>[[Operation_on_sets.mm|Flash]]</mm><br />
__FORCETOC__<br />
<br />
= Textbook =<br />
To add textbook links, please follow these instructions to: <br />
([{{fullurl:{{FULLPAGENAME}}/textbook|action=edit}} Click to create the subpage])<br />
<br />
=Additional Information=<br />
==Useful websites==<br />
==Reference Books==<br />
<br />
= Teaching Outlines =<br />
<br />
==Concept # 1- Union of sets==<br />
===Learning objectives===<br />
===Notes for teachers===<br />
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''<br />
<br />
===Activities===<br />
#Activity No #1<br />
#Activity No #2<br />
<br />
<br />
<br />
==Concept #2 - Intersection of Sets==<br />
===Learning objectives===<br />
#Elements can be shared across sets<br />
#Common elements can be combined<br />
===Notes for teachers===<br />
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''<br />
<br />
===Activities===<br />
#Activity No #1<br />
#Activity No #2<br />
<br />
==Concept #3 - Complement of Sets==<br />
===Learning objectives===<br />
#Elements can be shared across sets<br />
#Common elements can be combined<br />
===Notes for teachers===<br />
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''<br />
<br />
===Activities===<br />
#Activity No #1<br />
#Activity No #2<br />
<br />
==Concept #4 - Difference of Sets==<br />
===Learning objectives===<br />
#Elements can be shared across sets<br />
#Common elements can be combined<br />
===Notes for teachers===<br />
''These are short notes that the teacher wants to share about the concept, any locally relevant information, specific instructions on what kind of methodology used and common misconceptions/mistakes.''<br />
<br />
===Activities===<br />
#Activity No #1<br />
#Activity No #2<br />
<br />
=Assessment activities for CCE=<br />
{|class="wikitable"<br />
|-<br />
|ಕ್ರಮ ಸಂಖ್ಯೆ<br />
|ಪ್ರಶ್ನೆಗಳು<br />
|ಉತ್ತರಗಳು<br />
|ವೆನ್ ನಕ್ಷೆ <br />
|-<br />
|1<br />
|ಕಾಫೀ ಅಥವ ಟೀ ಯಾವು ದಾದರೂ <br> ಪಾನೀಯವನ್ನು ಸೇವಿಸು ವವರ ಸಂಖ್ಯೆ ಕಂಡು ಹಿಡಿಯಿರಿ.<br />
|ಸೂ ತ್ರ ಬಳಸಿರಿ<br />
|[[File:Venn1.png|150px]]<br />
|}<br />
[[File:activity1.odt]]<br />
<br />
= Hints for difficult problems =<br />
#Class 10 - Exercise 2.2 - Please click [http://karnatakaeducation.org.in/KOER/en/images/5/51/Solution_to_X_th_nproblems.pdf here for the solution]<br />
<br />
= Project Ideas =<br />
<br />
= Math Fun =<br />
<br />
'''Usage''' <br />
<br />
Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=File:Activity1.odt&diff=11850File:Activity1.odt2014-07-10T08:17:19Z<p>Naveenkumar: MsUpload</p>
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<div>MsUpload</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=File:Venn1.png&diff=11846File:Venn1.png2014-07-10T08:14:51Z<p>Naveenkumar: MsUpload</p>
<hr />
<div>MsUpload</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=File:Screenshot_from_2014-07-08_06-50-36.png&diff=11845File:Screenshot from 2014-07-08 06-50-36.png2014-07-10T08:13:09Z<p>Naveenkumar: MsUpload</p>
<hr />
<div>MsUpload</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=Real_numbers&diff=11533Real numbers2014-07-08T12:02:58Z<p>Naveenkumar: /* Additional Information */</p>
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<div><br />
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[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Philosophy Philosophy of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Pedagogy Teaching of Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Maths:_Curriculum_and_Syllabus Curriculum and Syllabus]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_Topics Topics in School Mathematics]<br />
|style=" width:10%; border:none; border-radius:5px;box-shadow: 10px 10px 10px #888888; background:#f9f9ff; vertical-align:middle; text-align:center; "|<br />
[http://karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]<br />
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= Concept Map =<br />
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= Textbook =<br />
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=Additional Information=<br />
==Useful websites==<br />
==Reference Books==<br />
{7} over {6}<br />
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= Teaching Outlines =<br />
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==Concept #==<br />
===Learning objectives===<br />
===Notes for teachers===<br />
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===Activity No # ===<br />
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==Concept #==<br />
===Learning objectives===<br />
===Notes for teachers===<br />
<br />
===Activity No # ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
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===Activity No # ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
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<br />
==Concept #==<br />
===Learning objectives===<br />
===Notes for teachers===<br />
===Activity No # ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
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*Question Corner<br />
===Activity No # ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
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*Question Corner<br />
===Activity No # ===<br />
{| style="height:10px; float:right; align:center;"<br />
|<div style="width:150px;border:none; border-radius:10px;box-shadow: 5px 5px 5px #888888; background:#f5f5f5; vertical-align:top; text-align:center; padding:5px;"><br />
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= Hints for difficult problems =<br />
Irrational numbers<br><br />
Exercise 1.3.1<br><br />
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1.Write four possible irrational numbers between 4 and 5.<br />
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Solution: <br />
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Consider the squares of 4 and 5<br />
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Square of 4 = 16 and Square of 5 = 25<br />
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We can also write 4 and 5 as<br />
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√ 16 =4 and √ 25 =5<br />
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Between and there exists<br />
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√ 17 ,√ 18 , √ 19 , √ 20 , √ 21 , √ 22 , √ 23 , √ 24<br />
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Use the following Geogebra applet to Visualise Rational and Irrational Numbers<br />
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= Project Ideas =<br />
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= Math Fun =<br />
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'''Usage''' <br />
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Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=File:Rectangle_.mm&diff=5316File:Rectangle .mm2013-09-04T10:25:12Z<p>Naveenkumar: Naveenkumar uploaded a new version of &quot;File:Rectangle .mm&quot;: MsUpload</p>
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<div>MsUpload</div>Naveenkumarhttps://karnatakaeducation.org.in/KOER/en/index.php?title=File:Rectangle_.mm&diff=5315File:Rectangle .mm2013-09-04T10:25:06Z<p>Naveenkumar: Naveenkumar uploaded a new version of &quot;File:Rectangle .mm&quot;: MsUpload</p>
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<div>MsUpload</div>Naveenkumar