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

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

Activity No # Construction of Transverse common tangent
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