The longest chord passes through the centre of the circle

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Textbook

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1. Karnataka text book for Class 10, Chapter 14 - Chord properties
2. Karnataka text book for Class 10, Chapter 15 - Tangent Properties

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 #4 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

Activities

1. Activity #1 - Cyclic quadrilateral
2. Activity #2 - Properties of cyclic quadrilateral

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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