Question

TP and TQ are tangents to a circle with centre O. prove that angle PTQ = 2 angle OPQ.

Answer 1

length of tangents drawn from an external point to a circle are equal.

so, TP=TQ

    angle TPQ = angle TQP

now PT is a tangent, and OP is radius,

therefore OP is perpendicular to PT

angle OPT = 90

angle OPQ + angle TPQ= 90

In triangle PTQ,

angle TPQ + angle TQP + angle PTQ = 180

angle TQP + angle TQP + angle PTQ = 180

2(angle TQP) + angle PTQ = 180

2(90- angle OPQ) + angle PTQ = 180

2(90) -2 angle OPQ + angle PTQ= 180

180- 2 OPQ +PTQ= 180

therefore,

           PTQ = 2 OPQ