Question

If the angle between two tangents drawn from an external point $P$ to a circle of radius a and a center $O$, is ${60}^o$ , then find the length of $OP$.

Answer 1

$Ap=BP$ ....(length of tangents from external point to circle are equal)

$\angle A=\angle B={90}^o$ .... (Tangent is $\perp$ to radius)

$OP=OP$ .... (common side)

$\therefore △AOP\cong △BOP$ ....(RHS test of congruence)

$\angle APO=\angle BPO={30}^o\to c.a.c.t$

$\angle AOP=\angle BOP={60}^o\to c.a.c.t$

$△AOP$ is ${30}^o-{60}^o-{90}^o$ triangle.

$\therefore △AOP$,

$\cos 60=\frac{OA}{OP}$

$OP=\frac{a}{\frac{1}{2}}=2a$