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

Concept #1 CHORD

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.]

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• Estimated Time
  

20 minutes

• Materials/ Resources needed:

Laptop, Geogebra file, projector and a pointer.

• 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

• 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.]

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• Estimated Time :40 minutes.
• Materials/ Resources needed:

Laptop, geogebra,projector and a pointer.

• 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 #

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

• Process/ Developmental Questions
• Evaluation
• Question Corner

Activity No #

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• 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

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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• 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.

Developmental Questions:

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 #

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Concept # Construction of tangent

Activity No # 1. Construction of tangent from an external point .

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Activity No # Construction of Direct common tangent

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Activity No # Construction of Transverse common tangent

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