Question

a tangent PT is drawn parallel to a chord AB of a circle.prove that APBis an isosceles triangle.

Answer 1

First you will consturct PO till D it is given in the attached figure with the help of that produce D

then produce OP_|_ TP
TP
||
AB

the ∠ ADP is 90 degrees 

PD is a bisector of AB which is triangle ADP ≃ triangle BDP 
so APB is an isosceles triangle

Answer 2

Join PO and produce to D. Due to which OP gets perpendicular to TP. 

Also, TP is parallel to AB 

∠ADP=90° (corresponding angles)

So, OD is perpendicular to AB. 

Hence, PD is a bisector of AB. 

that is AD = DB

In ΔADP and ΔBDP

AB =DB 

∠ADP=∠BDP 

 PD = DP  (common)

ΔADP= ΔBDP ( by SAS)

Hence, ΔPAD = ΔPBD (By CPCT)

Thus, APB is an isosceles triangle