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.

Activities

1. Activity No 1 - Theorem 1: Perpendicular bisector of a chord passes through the center of a circle
2. Activity No 2 - Theorem 2.Congruent chords are equidistant from the center of a circle

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.

Activities

1. Activity #1 - Understanding secant and tangent using Geogebra

Concept #3 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.

Activities

1. Activity #1 - Construction of Direct common tangent
2. Activity #2 - Construction of Transverse common tangent

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.

Hints for difficult problems

1. Tangents AP and AQ are drawn to circle with centre O, from an external point A. Prove that ∠PAQ=2.∠ OPQ

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