Question

If two tangents inclined at an angle ${60}^o$ are draw to a circle of radius $3cm,$ then find length of each tangent.

Answer 1

In $\Delta AOP$ & $\Delta BOP$,

$i)AO=OB=3cm$

$ii)AP=BP...........($ tangents drawn from an exterior point to the circle are equal$)$

$iii)OPiscommon$

$\therefore \Delta AOP\cong \Delta BOP$ ( By SSS congruence criterion )

$\Rightarrow \angle APO=\angle BPO=\frac{{60}^0}{2}={30}^0$

In $\Delta AOP,OA\perp AP......................($ radius is perpendicular to tangent at point of contact $)$

$\therefore \tan {30}^0=\frac{OA}{AP}=\frac{1}{\sqrt{3}}=\frac{3}{AP}$

$\Rightarrow AP=3\sqrt{3}cm$