Question

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

Answer 1

AB= AP
:. AB = CB ...
:. AB = CP........,,..(1)
(same angle but differentname)
:. AP=AC ........(2)
:. CP= CQ..... from (1),(2)

Answer 2

Answer with Step-by-step explanation:

In Δ APB

As we know sides opposite to equal angles are equal

∴ AB= AP ⇒ ∠ABP = ∠APB  

i.e.  ∠ABC = ∠APB                 -------(i)

Now

AC is chord , so Angles on the same segment are equal

∴   ∠ABC = ∠AQC  

i.e. ∠ABC =  ∠PQC                 --------(ii)

Vertically opposite angles are equal

∴ ∠ APB = ∠CPQ                      -------(iii)  

from (i) ,(ii),(iii) we get

∠PQC  = ∠CPQ

∴ CP= CQ ( sides opposite to equal angles are equal)

Hence proved