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

Teaching Outlines

Chord and its related theorems

Concept #1 Chord

Learning objectives

1. Meaning of circle and chord.
2. Method to measure the perpendicular distance of the chord from the centre of the circle.
3. Properties of chord.
4. Able to relate chord properties to find unknown measures in a circle.
5. Apply chord properties for proof of further theorems in circles.
6. 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. Basic concepts of a circle and its related terms should have been covered.

• Multimedia resources: Laptop.
• Website interactives/ links/ / Geogebra Applets:

This geogebra has been created by ITfc-Edu-team.

• Process:

1. Show the children the geogebra file and ask the listed questions below.

• 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 ?

• 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. Basics of circles and its related terms should have been done.

• Multimedia resources: Laptop, geogebra file, projector and a pointer.
• Website interactives/ links/ / Geogebra Applets : This geogebra file has been created by Tharanath Achar of Dakshina kannada.

• Process:

1. Show geogebra file and ask the questions below.

• 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 #

Click to Comment

• 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 # 1.Understanding Secant and Tangent using geogebra.

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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

This geogebra file has been made by ITfC-Edu-team

• 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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• 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 tangents

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 # 1. 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 : This geogebra file was created by Mallikarjun sudi of Yadgir.

• Process:

The teacher can explain the step by step construction of Direct common tangent and with an example.
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 ?
6. [Note for teachers : Evaluate if it is possible to construct a direct common tangent without the third circle.] Examine with the help of following geogebra file made by Ranjani.

Activity No # 2. Construction of Transverse common tangent

Click to Comment

• 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.

Developmental Questions

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 ?

Concept # Cyclic quadrilateral

Learning objectives

1. A quadrilateral ABCD is called cyclic if all four vertices of it lie on a circle.
2. In a cyclic quadrilateral the sum of opposite interior angles is 180 degrees.
3. If the sum of a pair of opposite angles of a quadrilateral is 180, the quadrilateral is cyclic.
4. In a cyclic quadrilateral the exterior angle is equal to interior opposite angle

Activity#1. Cyclic quadrilateral

• Estimated Time 10 minutes
• Materials/ Resources needed: Laptop, geogebra file, projector and a pointer.
• Prerequisites/Instructions, if any

1. Circles and quadrilaterals should have been covered.

• Multimedia resources : Laptop
• Website interactives/ links/ / Geogebra Applets; This geogebra file was created by ITfC-Edu-Team.

• Process:

1. Show the geogebra file.
2. Move points, the vertices of the quadrilateral and let the students observe the sum of opposite interior angles.

Developmental Questions:

1. What two figures do you see in the figure ?
2. Name the vertices of the quadrilateral.
3. Where are all the 4 vertices situated ?
4. Name the opposite interior angles of the quadrilateral.
5. What do you observe about them.

• Evaluation:

1. Compare the cyclic quadrilateral to circumcircle.

• Question Corner

1. Name this special quadrilateral.

Activity No # 2.Properties of a Cyclic quadrilateral

• Estimated Time: 45 minutes
• Materials/ Resources needed

coloured paper, pair if scissors, sketch pen, carbon paper, geometry box

• Prerequisites/Instructions, if any

1. Circles and quadrilaterals should have been covered.

• Multimedia resources
• Website interactives/ links/ / Geogebra Applets

This activity has been taken from the website http://mykhmsmathclass.blogspot.in/2007/11/class-ix-activity-16.html

• Process:

Note: Refer the above geogebra file to understand the below mentioned labelling.,br>

1. Draw a circle of any radius on a coloured paper and cut it.
2. Paste the circle cut out on a rectangular sheet of paper.
3. By paper folding get chords AB, BC, CD and DA in order.
4. Draw AB, BC, CD & DA. A cyclic quadrilateral ABCD is obtained.
5. Make a replica of cyclic quadrilateral ABCD using carbon paper.
6. Cut the replica into 4 parts such that each part contains one angle .
7. Draw a straight line on a paper.
8. Place angle BAD and angle BCD adjacent to each other on the straight line.Write the observation.
9. Place angle ABC and angle ADC adjacent to each other on the straight line . Write the observation.
10. Produce AB to form a ray AE such that exterior angle CBE is formed.
11. Make a replica of angle ADC and place it on angle CBE . Write the observation.

Developmental Questions:

1. How do you take radius ?
2. What is the circumference ?
3. What is a chord ?
4. What is a quadrilateral ?
5. Where are all four vertices of a quadrilateral located ?
6. What part are we trying to cut and compare ?
7. What can you infer ?

• Evaluation:

1. Angle BAD and angle BCD, when placed adjacent to each other on a straight line, completely cover the straight angle.What does this mean ?
2. Angle ABC and angle ADC, when placed adjacent to each other on a straight line, completely cover the straight angle.What can you conclude ?
3. Compare angle ADC with angle CBE.

• Question Corner:

Name the two properties of cyclic quarilaterals.

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