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The longest chord passes through the centre of the circle (view source)

Revision as of 21:47, 4 November 2019

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Investigating the diameter is the longest chord of a circle.
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Mathematics:_History The Story of Mathematics]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Text_Books#Mathematics_-_Textbooks Textbooks]
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[http://www.karnatakaeducation.org.in/KOER/en/index.php/Maths:_Question_Papers Question Bank]
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While creating a resource page, please click here for a resource creation [http://karnatakaeducation.org.in/KOER/en/index.php/Resource_Creation_Checklist '''checklist'''].
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= Concept Map =
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===Objectives===
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__FORCETOC__
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To understand longest chord passes through the centre and it is the diameter
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<mm>[[circles_and_lines.mm|flash]]</mm>
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===Estimated Time===
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30 minutes
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= Textbook =
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===Prerequisites/Instructions, prior preparations, if any===
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To add textbook links, please follow these instructions to:
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Prior knowledge of point, lines, angles, polygons
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([{{fullurl:{{FULLPAGENAME}}/textbook|action=edit}} Click to create the subpage])
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=Additional Information=
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===Materials/ Resources needed===
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==Useful websites==
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* Digital : Computer, geogebra application, projector.
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#[http://www.regentsprep.org/Regents/math/geometry/GP14/PracCircleSegments.htm www.regentsprep.com] conatins good objective problems on chords and secants <br>
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* Non digital : Worksheet and pencil, compass, strings
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#[http://www.mathwarehouse.com/geometry/circle/tangents-secants-arcs-angles.php www.mathwarehouse.com] contains good content on circles for different classes<br>
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* Geogebra files :  [https://ggbm.at/c4eg7q2u Diameter is longest chord.ggb]
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#[http://staff.argyll.epsb.ca/jreed/math20p/circles/tangent.htm staff.argyll]  contains good simulations
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{{Geogebra|c4eg7q2u}}
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==Reference Books==
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===Process (How to do the activity)===
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Use the geogebra file to show how diameter is the longest chord.
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= Teaching Outlines
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Move the points on the circle to show the changes in the triangle.
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Chord and its related theorems
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==Concept #1 CHORD==
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===Learning objectives===
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The students should be able to:
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# Recall the meaning of circle and chord.
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# They should know the method to measure the perpendicular distance of the chord from the centre of the circle.
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# State Properties of chord.
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# By studying the theorems related to chords, the students should know that a chord in a circle is an important concept .
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# They should be able to relate chord properties to find unknown measures in a circle.
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# They should be able to apply chord properties for proof of further theorems in circles.
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# The students should  understand the meaning of congruent chords.
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===Notes for teachers===
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What is the condition with respect to sides for formation of a triangle. Sum of two sides is larger than the third side.
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# A chord is a straight line joining 2 points on the circumference of a circle.
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# Chords within a circle can be related in many ways.  
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# The theorems that involve chords of a circle are :
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* Perpendicular bisector of a chord passes through the center of a circle.
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* Congruent chords are equidistant from the center of a circle.
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* If two chords in a circle are congruent, then their intercepted arcs are congruent.
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* If two chords in a circle are congruent, then they determine two central angles that are congruent.
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===Activity No 1[Theorem 1: Perpendicular bisector of a chord passes through the center of a circle.] ===
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Compare the chord length with sum of two radii. When is the triangle reduced to a line segment.
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{| style="height:10px; float:right; align:center;"
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|<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;">
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''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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|}
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*Estimated Time <br>
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20 minutes
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*Materials/ Resources needed:
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Laptop, Geogebra file, projector and a pointer.
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*Prerequisites/Instructions, if any
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# The students should know the basic concepts of a circle and its related terms.
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# They should have prior knowledge of chord and construction of perpendicular bisector to the chord.
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*Multimedia resources: Laptop
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*Website interactives/ links/ / Geogebra Applets
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What can you conclude about the chord? When is it the largest?
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />
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*Process:
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# Show the children the geogebra file.
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# Let them identify the chord. Ask them to define a chord.
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# Let them recall what a perpendicular bisector is.
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# Show them the second chord.
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# Let students observe if everytime the perpendicular bisector of the chord passes through the centre of the circle.
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*Developmental Questions:
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# What is a chord ?
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# At how many points on the circumference does the chord touch a circle .
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# What is a bisector ?
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# What is a perpendicular bisector ?
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# In each case the perpendicular bisector passes through which point ?
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# Can anyone explain why does the perpendicular bisector always passes through the centre of the circle ?
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*Evaluation
+
[[Category:Circles]]
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# What is the angle formed at the point of intersection of chord and radius ?
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# Are the students able to understand what a perpendicular bisector is ?
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# Are the students realising that perpendicular bisector drawn for any length of chords for any circle always passes through the center of the circle .
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*Question Corner:
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# What do you infer ?
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# How can you reason that the perpendicular bisector for any length of chord always passes through the centre of the circle.
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===Activity No # 2.[Theorem 2.Congruent chords are equidistant from the center of a circle.] ===
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{| style="height:10px; float:right; align:center;"
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|<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;">
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''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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|}
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*Estimated Time :40 minutes.
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*Materials/ Resources needed:
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Laptop, geogebra,projector and a pointer.
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*Prerequisites/Instructions, if any
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# The students should have prior knowledge of a circle, its centre, radius, circumference and a    chord.
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# They should know that the length of the chord means its perpendicular distance from the centre.
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# They should know to draw perpendicular bisector to a given chord.
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# They should know the meaning of the term congruent and equidistant.
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*Multimedia resources: Laptop, geogebra file, projector and a pointer.
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*Website interactives/ links/ / Geogebra Applets
  −
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*Process:
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# The teacher can reiterate the prior knowledge on circles.
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# Revise the procedure of drawing chords of given length accurately in a circle.
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# Revise what congruent chords mean.
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# Show geogebra file and explain to help them understand the theorem.
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*Developmental Questions:
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# What is a chord ?
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# Name the centre of the circle.
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# How do you draw congruent chords in a circle ?
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# How many chords do you see in the figure ? Name them.
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# If  both the chords are congruent, what can you say about the length of both the chords ?
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# How can we measure the length of the chord ?
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# What is the procedure to draw perpendicular bisector ?
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# What does theorem 1 say ? Do you all remember ?
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# What is the length of both chords here ?
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# What can you conclude ?
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# Repeat this for circles of different radii and for different lengths of congruent chords.
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*Evaluation:
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# Were the students able to comprehend the drawing of congruent chords in a circle ?
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# Were the students able to comprehend why congruent chords are always equal for a given circle. Let any student explain the analogy.
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# Are the students able to understand that this theorem can be very useful in solving problems related to circles and triangles ?
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*Question Corner:
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# What is a chord ?
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# What are congruent chords ?
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# Why do you think congruent chords are always equal for a circle of given radius ?
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===Activity No # ===
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{| style="height:10px; float:right; align:center;"
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|<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;">
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''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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|}
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*Estimated Time
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*Materials/ Resources needed
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*Prerequisites/Instructions, if any
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*Multimedia resources
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*Website interactives/ links/ / Geogebra Applets
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*Process/ Developmental Questions
  −
*Evaluation
  −
*Question Corner
  −
 
  −
===Activity No # ===
  −
{| style="height:10px; float:right; align:center;"
  −
|<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;">
  −
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
  −
|}
  −
*Estimated Time
  −
*Materials/ Resources needed
  −
*Prerequisites/Instructions, if any
  −
*Multimedia resources
  −
*Website interactives/ links/ / Geogebra Applets
  −
 
  −
*Process/ Developmental Questions
  −
*Evaluation
  −
*Question Corner
  −
 
  −
==Concept #2.Secant and Tangent==
  −
===Learning objectives===
  −
# The secant is a line passing through a circle touching it at any two points on the circumference.
  −
# A tangent is a line toucing the circle at only one point on the circumference.
  −
===Notes for teachers===
  −
===Activity No # ===
  −
{| style="height:10px; float:right; align:center;"
  −
|<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;">
  −
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
  −
|}
  −
*Estimated Time: 15 minutes
  −
*Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
  −
*Prerequisites/Instructions, if any:
  −
# The students should have a prior knowledge about a circle and its basic parts and terms.
  −
# They should know the clear distinction between radius, diameter, chord, secant and tangent.
  −
*Multimedia resources : Laptop and projector
  −
*Website interactives/ links/ / Geogebra Applets
  −
<ggb_applet width="1282" height="601" version="4.0" ggbBase64="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" enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />
  −
*Process:
  −
# The teacher can show the geogebra file.
  −
# Move the points on circumference and explain secant.
  −
# When both endpoints of secant meet, it becomes a tangent.
  −
Developmental Questions:
  −
# Name the points on the circumference of the circle.
  −
# At how many points is the line touching the circle ?
  −
# What is the line called ?
  −
*Evaluation
  −
# What is the difference between the secant and a tangent?
  −
# What is the difference between the chord and a secant ?
  −
*Question Corner
  −
# Can you draw a secant touching 3 points on the circle ?
  −
# At how many points does a tangent touch a circle ?
  −
# How many tangents can be drawn to a circle ?
  −
# How many tangents can be drawn to a circle at any one given point ?
  −
# How many parallel tangents can a circle have at the most ?
  −
 
  −
===Activity No # ===
  −
{| style="height:10px; float:right; align:center;"
  −
|<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;">
  −
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
  −
|}
  −
*Estimated Time
  −
*Materials/ Resources needed
  −
*Prerequisites/Instructions, if any
  −
*Multimedia resources
  −
*Website interactives/ links/ / Geogebra Applets
  −
*Process/ Developmental Questions
  −
*Evaluation
  −
*Question Corner
  −
 
  −
==Concept # Construction of tangent==
  −
===Learning objectives===
  −
# The students should know that tangent is a straight line touching the circle at one and only point.
  −
# They should understand that a tangent is perpendicular to the radius of the circle.
  −
# The construction protocol of a tangent.
  −
# Constructing a tangent to a point on the circle.
  −
# Constructing tangents to a circle from external point at a given distance.
  −
# A tangent that is common to two circles is called a common tangent.
  −
# A common tangent with both centres on the same side of the tangent is called a direct common tangent.
  −
# A common tangent with both centres on either side of the tangent is called a transverse common tangent.
  −
 
  −
===Notes for teachers===
  −
===Activity No # Construction of Direct common tangent ===
  −
{| style="height:10px; float:right; align:center;"
  −
|<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;">
  −
''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
  −
|}
  −
*Estimated Time: 90 minutes
  −
*Materials/ Resources needed:
  −
# Laptop, geogebra file, projector and a pointer.
  −
# Students' individual construction materials.
  −
*Prerequisites/Instructions, if any
  −
# The students should have prior knowledge of a circle , tangent and the limiting case of a secant as a tangent.
  −
# They should understand that a tangent is always perpendicular to the radius of the circle.
  −
# They should know construction of a tangent to a given point.
  −
# If the same straight line is a tangent to two or more circles, then it is called a common tangent.
  −
# If the centres of the circles lie on the same side of the common tangent, then the tangent is called a direct common tangent.
  −
# Note: In general,
  −
*The two circles are named as C1 and C2
  −
* The distance between the centre of two circles is 'd'
  −
* Radius of one circle is taken as 'R' and other as 'r'
  −
* The length of tangent is 't'
  −
*Multimedia resources:Laptop
  −
*Website interactives/ links/ / Geogebra Applets
  −
<ggb_applet width="1280" height="600" version="4.0" 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" enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />
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*Process:
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The teacher can explain the step by step construction of Direct common tangent  and with an example.
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[Note for  teachers : Evaluate if it is possible to construct a direct common tangent without the third circle.]
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Developmental Questions:
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#What is a tangent
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# What is a common tangent ?
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# What is a direct common tangent ?
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# What is R and r  ?
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# What does the length OA represent here ?
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# Why was a third circle constructed ?
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# Let us try to construct direct common tangent without the third circle and see.
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# What should be the radius of the third circle ?
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# Why was OA bisected and semi circle constructed ?
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# What were OB and OC extended ?
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# What can you say about lines AB and AC ?
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# Name the direct common tangents .
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# At what points is the tangent touching the circles ?
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# Identify the two right angled triangles formed from the figure ? What do you understand ?
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*Evaluation:
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# Is the student able to comprehend the sequence of steps in constructing the tangent.
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# Is the student able to identify error areas while constructing ?
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# Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
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# Is the student able to appreciate that the direct common tangents from the same external point are equal and subtend equal angles at the center.
  −
*Question Corner:
  −
# What do you think are the applications of tangent constructions ?
  −
# What is the formula to find the length of direct common tangent ?
  −
# Can a direct common tangent be drawn to two circles one inside the other ? 
  −
# Observe the point of intersection of extended tangents in relation with the centres of two circles. Infer.
  −
# What are properties of direct common tangents ?
  −
 
  −
===Activity No # Construction of Transverse common tangent===
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{| style="height:10px; float:right; align:center;"
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|<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;">
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''[http://www.karnatakaeducation.org.in/?q=node/305 Click to Comment]''</div>
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|}
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*Estimated Time: 45 minutes
  −
*Materials/ Resources needed:
  −
# Laptop, geogebra file, projector and a pointer.
  −
# Students' individual construction materials.
  −
*Prerequisites/Instructions, if any
  −
# The students should have prior knowledge of a circle , tangent and direct and transverse common tangents .
  −
# They should understand that a tangent is always perpendicular to the radius of the circle.
  −
# They should know construction of a tangent to a given point.
  −
# If the same straight line is a tangent to two or more circles, then it is called a common tangent.
  −
# If the centres of the circles lie on opposite side of the common tangent, then the tangent is called a transverse common tangent.
  −
# Note: In general,
  −
*The two circles are named as C1 and C2
  −
* The distance between the centre of two circles is 'd'
  −
* Radius of one circle is taken as 'R' and other as 'r'
  −
* The length of tangent is 't'
  −
*Multimedia resources: Laptop
  −
*Website interactives/ links/ / Geogebra Applets
  −
<ggb_applet width="1280" height="600" version="4.0" 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*Process:
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# The teacher can explain the step by step construction of Transverse common tangent.
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Developmental Questions
  −
# What is a transverse common tangent ?
  −
# What is the radius of the third circle ?
  −
# What is the difference in finding the radius of the third circle in constructing Dct and that of Tct ?
  −
# Why was a third circle constructed ?
  −
# Let us try to construct transverse common tangent without the third circle and see.
  −
# Name the transverse common tangents .
  −
# At what points is the tangent touching the circles ?
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*Evaluation:
  −
# Is the student able to comprehend the sequence of steps in constructing the tangent.
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# Is the student able to identify error areas while constructing ?
  −
# Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
  −
# Is the student able to understand the difference in the construction protocol between direct common tangent and transverse common tangent ?
  −
*Question Corner:# What do you think are the applications of tangent constructions ?
  −
# What is the formula to find the length of transverse common tangent ?
  −
# Can a direct common tangent be drawn to two circles one inside the other ? 
  −
# What are properties of transverse common tangents ?
  −
*Evaluation:
  −
# Were the students able to comprehend the steps in transverse common tangent construction ?
  −
 
  −
*Question Corner:
  −
# Can you construct a transverse common tangent without the third circle ?
  −
 
  −
==Concept # Cyclic quadrilateral==
  −
===Learning objectives===
  −
# The students should learn that a quadrilateral ABCD is called cyclic if all four vertices of it lie on a circle.
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# The sum of either pair of opposite angles of a cyclic quadrilateral is 180 degrees.
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# If the sum of a pair of opposite angles of a quadrilateral is 180, the quadrilateral is cyclic.
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===Notes for teachers===
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*Activity No # Construction of cyclic quadrilateral                                                                                                               
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*Estimated Time
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*Materials/ Resources needed
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*Prerequisites/Instructions, if any
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*Multimedia resources
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*Website interactives/ links/ / Geogebra Applets
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*Process/ Developmental Questions
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*Evaluation
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*Question Corner
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*Activity No #                                                                                                             
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*Estimated Time
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*Materials/ Resources needed
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*Prerequisites/Instructions, if any
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*Multimedia resources
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*Website interactives/ links/ / Geogebra Applets
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*Process/ Developmental Questions
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*Evaluation
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*Question Corner
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= Hints for difficult problems =
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= Project Ideas =
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= Math Fun =
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'''Usage'''
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Create a new page and type <nowiki>{{subst:Math-Content}}</nowiki> to use this template
 
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