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prove that equal chords of a cycle subtend equal angle at centre​

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Mathematics

Grade 11

Congruent Circles

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Prove that the equal chords of two congruent circles subtend equal angles at their respective centres.

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Hint: Use the fact that the radii of two congruent circles are equal and hence prove that OA = O’A’ and OB = O’B’. Use the fact that since the chords are equal AB = A’B’ and hence prove that the triangle ABC and A’B’C’ are congruent and hence prove that ∠AOC=∠A′O′C′. Hence prove that the equal chords of congruent circles subtend equal angles at the centres of their corresponding circles.

Complete step by step answer:

Given: Two circles with centre O and O’ have equal radii. AB is the chord of the circle with centre O and A’B’ is a chord of the circle with centre O’.

To prove ∠AOB=∠A′O′B′

Proof:

Since the circle have equal radii, we have

OA = O’A’ and OB = O’B’

Now, in triangle AOB and A’O’B’, we have

AO = A’O’ (proved above)

OB = O’B’ (Proved above)

AB = A’B’ (Given).

Hence by S.S.S congruence criterion, we have

ΔAOB≅ΔA′O′B′

Hence, we have

∠AOB=∠A′O′B′ (Corresponding parts of congruent triangles)

Hence, equal chords of two congruent circles subtend equal angles at their respective centres.

Hence, proved.