CBSE Class 10 Answered

Let OP meets the circle at a point Q.

It  is given that OP = 2r

So, OP OP = QP + OP

QP = r.

Therfore, Q is the mid point of OP.

Now AP is tnagent at A and OA is radius

OA is perpendicular to AP.

Therefore, Triangle OAP is right triangle at A . Q is the mid point of hypotnuese OP.

Therfore, AQ = OQ = OA

i.e Triangle OAQ is equilateral.

So, AOQ = 60^o

Therefore, APO = 180^o - 90^o- 60^o  = 30^o

APB = 2APO  = 60^o

Also, PA = PB

PAB = PBA

Now in triangle APB, we have

PAB  + PBA + APB = 180^o 

PAB = 60^o

So, triangle APB is an equilateral triangle.