Question

In the given figure. P is any point on the chord BC of a circle such that AB = AP. Prove that CP = CQ.

In Fig., P is a point on the chord BC such that AB = AP. Prove that: CP = CQ.

Answer 1

Given:

P is a point on the chord BC such that AB = AP.

To Prove:

1. In ΔABP, AB = AP, so

∠ABP = ∠BPA...(1)

Chord AC subtends equal angles at B, P, Q:

∠ABP = ∠ACP          ...(2)

∠BPA = ∠CPQ           ...(3)

From (1), (2), (3):

∠ACP = ∠CPQ

So triangle CPQ is isosceles:

CP = CQ