Q 21 in the figure,p is any point on the chord BC of a circle such that AB=AP.Prove that CP=CQ

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

TPT: CP = CQ

proof:

in the triangle ABP, AB = AP

therefore ∠ABP = ∠APB........(1)

since angle subtended by the chord at the circumference are equal.

∠ABC = ∠AQC  [angles subtended by the chord AC at circumference ]

∠ABP = ∠AQC ..........(2) [∠ABC = ∠ABP same angle]

∠APB = ∠CPQ ......(3)

from (1), (2) and (3):

∠CPQ = ∠AQC i.e. ∠CPQ = ∠PQC 

therefore CP = CQ [sides opposite to equal angles are equal]

hope this helps you.