The longest chord passes through the centre of the circle

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=Additional Information=

Useful websites

 * 1) www.regentsprep.com conatins good objective problems on chords and secants
 * 2) www.mathwarehouse.com contains good content on circles for different classes
 * 3) staff.argyll contains good simulations

Reference Books
= Teaching Outlines Chord and its related theorems

Learning objectives
The students should be able to:
 * 1) Recall the meaning of circle and chord.
 * 2) They should know the method to measure the perpendicular distance of the chord from the centre of the circle.
 * 3) State Properties of chord.
 * 4) By studying the theorems related to chords, the students should know that a chord in a circle is an important concept.
 * 5) They should be able to relate chord properties to find unknown measures in a circle.
 * 6) They should be able to apply chord properties for proof of further theorems in circles.
 * 7) The students should  understand the meaning of congruent chords.

Notes for teachers

 * 1) A chord is a straight line joining 2 points on the circumference of a circle.
 * 2) Chords within a circle can be related in many ways.
 * 3) The theorems that involve chords of a circle are :
 * Perpendicular bisector of a chord passes through the center of a circle.
 * Congruent chords are equidistant from the center of a circle.
 * If two chords in a circle are congruent, then their intercepted arcs are congruent.
 * If two chords in a circle are congruent, then they determine two central angles that are congruent.

Activity No 1[Theorem 1: Perpendicular bisector of a chord passes through the center of a circle.]
20 minutes Laptop, Geogebra file, projector and a pointer.
 * Estimated Time
 * Materials/ Resources needed:
 * Prerequisites/Instructions, if any
 * 1) The students should know the basic concepts of a circle and its related terms.
 * 2) They should have prior knowledge of chord and construction of perpendicular bisector to the chord.
 * Multimedia resources: Laptop

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 * Website interactives/ links/ / Geogebra Applets
 * Process:
 * 1) Show the children the geogebra file.
 * 2) Let them identify the chord. Ask them to define a chord.
 * 3) Let them recall what a perpendicular bisector is.
 * 4) Show them the second chord.
 * 5) Let students observe if everytime the perpendicular bisector of the chord passes through the centre of the circle.
 * Developmental Questions:
 * 1) What is a chord ?
 * 2) At how many points on the circumference does the chord touch a circle.
 * 3) What is a bisector ?
 * 4) What is a perpendicular bisector ?
 * 5) In each case the perpendicular bisector passes through which point ?
 * 6) Can anyone explain why does the perpendicular bisector always passes through the centre of the circle ?


 * Evaluation
 * 1) What is the angle formed at the point of intersection of chord and radius ?
 * 2) Are the students able to understand what a perpendicular bisector is ?
 * 3) Are the students realising that perpendicular bisector drawn for any length of chords for any circle always passes through the center of the circle.
 * Question Corner:
 * 1) What do you infer ?
 * 2) How can you reason that the perpendicular bisector for any length of chord always passes through the centre of the circle.

Activity No # 2.[Theorem 2.Congruent chords are equidistant from the center of a circle.]
Laptop, geogebra,projector and a pointer. <ggb_applet width="1280" height="600" version="4.0" 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 * Estimated Time :40 minutes.
 * Materials/ Resources needed:
 * Prerequisites/Instructions, if any
 * 1) The students should have prior knowledge of a circle, its centre, radius, circumference and a     chord.
 * 2) They should know that the length of the chord means its perpendicular distance from the centre.
 * 3) They should know to draw perpendicular bisector to a given chord.
 * 4) They should know the meaning of the term congruent and equidistant.
 * Multimedia resources: Laptop, geogebra file, projector and a pointer.
 * Website interactives/ links/ / Geogebra Applets
 * Process:
 * 1) The teacher can reiterate the prior knowledge on circles.
 * 2) Revise the procedure of drawing chords of given length accurately in a circle.
 * 3) Revise what congruent chords mean.
 * 4) Show geogebra file and explain to help them understand the theorem.
 * Developmental Questions:
 * 1) What is a chord ?
 * 2) Name the centre of the circle.
 * 3) How do you draw congruent chords in a circle ?
 * 4) How many chords do you see in the figure ? Name them.
 * 5) If  both the chords are congruent, what can you say about the length of both the chords ?
 * 6) How can we measure the length of the chord ?
 * 7) What is the procedure to draw perpendicular bisector ?
 * 8) What does theorem 1 say ? Do you all remember ?
 * 9) What is the length of both chords here ?
 * 10) What can you conclude ?
 * 11) Repeat this for circles of different radii and for different lengths of congruent chords.
 * Evaluation:
 * 1) Were the students able to comprehend the drawing of congruent chords in a circle ?
 * 2) Were the students able to comprehend why congruent chords are always equal for a given circle. Let any student explain the analogy.
 * 3) Are the students able to understand that this theorem can be very useful in solving problems related to circles and triangles ?
 * Question Corner:
 * 1) What is a chord ?
 * 2) What are congruent chords ?
 * 3) Why do you think congruent chords are always equal for a circle of given radius ?

Activity No #

 * Estimated Time
 * Materials/ Resources needed
 * Prerequisites/Instructions, if any
 * Multimedia resources
 * Website interactives/ links/ / Geogebra Applets


 * Process/ Developmental Questions
 * Evaluation
 * Question Corner

Activity No #

 * Estimated Time
 * Materials/ Resources needed
 * Prerequisites/Instructions, if any
 * Multimedia resources
 * Website interactives/ links/ / Geogebra Applets


 * Process/ Developmental Questions
 * Evaluation
 * Question Corner

Learning objectives

 * 1) The secant is a line passing through a circle touching it at any two points on the circumference.
 * 2) A tangent is a line toucing the circle at only one point on the circumference.

Activity No #
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> Developmental Questions:
 * Estimated Time: 15 minutes
 * Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
 * Prerequisites/Instructions, if any:
 * 1) The students should have a prior knowledge about a circle and its basic parts and terms.
 * 2) They should know the clear distinction between radius, diameter, chord, secant and tangent.
 * Multimedia resources : Laptop and projector
 * Website interactives/ links/ / Geogebra Applets
 * Process:
 * 1) The teacher can show the geogebra file.
 * 2) Move the points on circumference and explain secant.
 * 3) When both endpoints of secant meet, it becomes a tangent.
 * 1) Name the points on the circumference of the circle.
 * 2) At how many points is the line touching the circle ?
 * 3) What is the line called ?
 * Evaluation
 * 1) What is the difference between the secant and a tangent?
 * 2) What is the difference between the chord and a secant ?
 * Question Corner
 * 1) Can you draw a secant touching 3 points on the circle ?
 * 2) At how many points does a tangent touch a circle ?
 * 3) How many tangents can be drawn to a circle ?
 * 4) How many tangents can be drawn to a circle at any one given point ?
 * 5) How many parallel tangents can a circle have at the most ?

Activity No #

 * Estimated Time
 * Materials/ Resources needed
 * Prerequisites/Instructions, if any
 * Multimedia resources
 * Website interactives/ links/ / Geogebra Applets
 * Process/ Developmental Questions
 * Evaluation
 * Question Corner

Learning objectives

 * 1) The students should know that tangent is a straight line touching the circle at one and only point.
 * 2) They should understand that a tangent is perpendicular to the radius of the circle.
 * 3) The construction protocol of a tangent.
 * 4) Constructing a tangent to a point on the circle.
 * 5) Constructing tangents to a circle from external point at a given distance.
 * 6) A tangent that is common to two circles is called a common tangent.
 * 7) A common tangent with both centres on the same side of the tangent is called a direct common tangent.
 * 8) A common tangent with both centres on either side of the tangent is called a transverse common tangent.

Activity No # Construction of Direct common tangent
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" />
 * Estimated Time: 90 minutes
 * Materials/ Resources needed:
 * 1) Laptop, geogebra file, projector and a pointer.
 * 2) Students' individual construction materials.
 * Prerequisites/Instructions, if any
 * 1) The students should have prior knowledge of a circle, tangent and the limiting case of a secant as a tangent.
 * 2) They should understand that a tangent is always perpendicular to the radius of the circle.
 * 3) They should know construction of a tangent to a given point.
 * 4) If the same straight line is a tangent to two or more circles, then it is called a common tangent.
 * 5) If the centres of the circles lie on the same side of the common tangent, then the tangent is called a direct common tangent.
 * 6) 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

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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> The teacher can explain the step by step construction of Direct common tangent and with an example. [Note for teachers : Evaluate if it is possible to construct a direct common tangent without the third circle.] Developmental Questions:
 * Process:
 * 1) What is a tangent
 * 2) What is a common tangent ?
 * 3) What is a direct common tangent ?
 * 4) What is R and r  ?
 * 5) What does the length OA represent here ?
 * 6) Why was a third circle constructed ?
 * 7) Let us try to construct direct common tangent without the third circle and see.
 * 8) What should be the radius of the third circle ?
 * 9) Why was OA bisected and semi circle constructed ?
 * 10) What were OB and OC extended ?
 * 11) What can you say about lines AB and AC ?
 * 12) Name the direct common tangents.
 * 13) At what points is the tangent touching the circles ?
 * 14) Identify the two right angled triangles formed from the figure ? What do you understand ?
 * Evaluation:
 * 1) Is the student able to comprehend the sequence of steps in constructing the tangent.
 * 2) Is the student able to identify error areas while constructing ?
 * 3) Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
 * 4) 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:
 * 1) What do you think are the applications of tangent constructions ?
 * 2) What is the formula to find the length of direct common tangent ?
 * 3) Can a direct common tangent be drawn to two circles one inside the other ?
 * 4) Observe the point of intersection of extended tangents in relation with the centres of two circles. Infer.
 * 5) What are properties of direct common tangents ?

Activity No # Construction of Transverse common tangent
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enableRightClick="false" showAlgebraInput="false" enableShiftDragZoom="true" showMenuBar="false" showToolBar="false" showToolBarHelp="true" enableLabelDrags="false" showResetIcon="true" /> Developmental Questions
 * Estimated Time: 45 minutes
 * Materials/ Resources needed:
 * 1) Laptop, geogebra file, projector and a pointer.
 * 2) Students' individual construction materials.
 * Prerequisites/Instructions, if any
 * 1) The students should have prior knowledge of a circle, tangent and direct and transverse common tangents.
 * 2) They should understand that a tangent is always perpendicular to the radius of the circle.
 * 3) They should know construction of a tangent to a given point.
 * 4) If the same straight line is a tangent to two or more circles, then it is called a common tangent.
 * 5) If the centres of the circles lie on opposite side of the common tangent, then the tangent is called a transverse common tangent.
 * 6) 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
 * Process:
 * 1) The teacher can explain the step by step construction of Transverse common tangent.
 * 1) What is a transverse common tangent ?
 * 2) What is the radius of the third circle ?
 * 3) What is the difference in finding the radius of the third circle in constructing Dct and that of Tct ?
 * 4) Why was a third circle constructed ?
 * 5) Let us try to construct transverse common tangent without the third circle and see.
 * 6) Name the transverse common tangents.
 * 7) At what points is the tangent touching the circles ?
 * Evaluation:
 * 1) Is the student able to comprehend the sequence of steps in constructing the tangent.
 * 2) Is the student able to identify error areas while constructing ?
 * 3) Is the student observing that the angle between the tangent and radius at the point of intersection is 90º ?
 * 4) 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 ?
 * 1) What is the formula to find the length of transverse common tangent ?
 * 2) Can a direct common tangent be drawn to two circles one inside the other ?
 * 3) What are properties of transverse common tangents ?
 * Evaluation:
 * 1) Were the students able to comprehend the steps in transverse common tangent construction ?


 * Question Corner:
 * 1) Can you construct a transverse common tangent without the third circle ?

= Hints for difficult problems =

= Project Ideas =

= Math Fun =

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