Prove that a diameter of a circle which bisects a chord of the circle

1.	Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.
Answer» Given that, PQ is a diameter of circle which bisects chord AB to C To prove : PQ bisects ∠AOB Proof : In ΔAOC and ΔBOC, OA = OB (Radius of circle) OC = OC (Common) AC = BC (Given) Then, Δ ADC ≅ Δ BOC (By SSS congruence rule) ∠AOC = ∠BOC (By c.p.c.t) Hence, PQ bisects ∠AOB.

Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.

Answer»

Given that,

PQ is a diameter of circle which bisects chord AB to C

To prove :

PQ bisects ∠AOB

Proof :

In ΔAOC and ΔBOC,

OA = OB (Radius of circle)

OC = OC (Common)

AC = BC (Given)

Then,

Δ ADC ≅ Δ BOC

(By SSS congruence rule)

∠AOC = ∠BOC

(By c.p.c.t)

Hence,

PQ bisects ∠AOB.

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Prove that a diameter of a circle which bisects a chord of the circle also bisects the angle subtended by the chord at the centre of the circle.

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