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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• Evaluation
• Question Corner

Concept # Construction of tangent

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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• 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

• Process:

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:

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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• Estimated Time
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• Process/ Developmental Questions
• Evaluation
• Question Corner

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