Question

In figure P is a point on the chord BC such that AB=AP prove that CP=CQ​

Answer 1

Answer:

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Answer 2

Answer:

Hence proved.

Step-by-step explanation:

We have to prove that CP = CQ i.e., Δ CPQ is an isosceles triangle. for this it is sufficient to prove that ∠ CPQ = ∠ CQP.

In Δ ABP, we have

AB = AP

⇒ ∠ APB = ∠ ABP

⇒  ∠ CPQ = ∠ ABP      ...(i)( ∵ ∠APB and ∠ CPQ are vertically opposite angles ∴ ∠APB = ∠ CPQ )

Now consider arc AC. Clearly, it subtends ∠ABC and ∠AQC at points B and Q.

∴ ∠ABC = ∠AQC      ...( ∵ Angles in the same segment)

⇒ ∠ABP = ∠PQC      ...( ∵∠ ABC = ∠ ABP and ∠AQC = ∠PQC )

⇒ ∠ABP = ∠CQP      ....(ii)( ∵ ∠PQC = ∠CQP )

From (i) and (ii), we get

∠ CPQ = ∠CQP

⇒ CQ = CP

Hence proved.