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Properties of chords

Properties of chords. PREPARED BY P.E.VENUGOPALAN, K.V.KELTRON NAGAR. Theorem. Equal chords of a circle subtend equal angles at the centre. D. A. O. B. C. Proof. Given : we are given that chords AB and CD of a circle with centre O are equal To prove : ∟AOB =∟ COD Proof :

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Properties of chords

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  1. Properties of chords PREPARED BY P.E.VENUGOPALAN, K.V.KELTRON NAGAR

  2. Theorem Equal chords of a circle subtend equal angles at the centre D A O B C

  3. Proof Given : we are given that chords AB and CD of a circle with centre O are equal To prove: ∟AOB =∟ COD Proof: In triangles AOB and COD OA=OC (radii of the circle) OB=OD (radii of the circle) AB = CD (given) ΔAOB ≡ ΔCOD (SSS) This gives ∟AOB =∟ COD (CPCT)-proved

  4. (Theorem 10.2) • If the angles subtended by the chords of a circle at the centre are equal ,then the chords are equal Activity: Draw a circle of radius 4cm. Mark two points on the circle .join them to the centre. Measure the central angle. Mark another pair of points such that the angle between the radii through these points is equal to the measure got above . Join the pair of points in each case. Measure the length of the corresponding chords. What do you observe?. Can you verify it geometrically ? Try !

  5. (Theorem10.3) • The perpendicular from the centre of a circle to a chord bisects the chord. Activity : join the centre to the endpoints of the chord to get two triangles. Name them appropriately and try to prove the theorem O

  6. (Theorem10.4) • The line drawn through the centre of a circle to bisect a chord is perpendicular to the chord Activity : Draw any chord of a circle and locate its midpoint. Join the midpoint to the centre of the circle and draw the line passing through these points. measure the angle(s) made by this line with the chord. what do you observe? Is it true for any chord of the circle? Justify your answer.

  7. Equal chords • Equal chords of a circle are equidistant from the centre Activity : Draw a circle of any radius. Draw a pair of equal chords. locate their midpoints. Join the midpoints to the centre of the circle. Measure distance from the centre to the midpoint. Are they same? Is this true for a pair of congruent circles? Verify practically. On the basis of your observation re-write the modified statement of the above theorem

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