Question

If two tangents inclined at an angle ${60}^{\circ }$ are drawn to a circle of radius $3cm,$ then the length of each tangent is
(A) $\frac{3}{2}$$\sqrt{3}cm$
(B) $6cm$
(C) $3cm$
(D) 3 $\sqrt{3}cm$

Answer 1

Let $P$ be an external point and a pair of tangents is drawn from point P and angle between these two tangents is ${60}^{\circ }$

Radius of the circle $=3cm$
Join OA and OP
Also, OP is a bisector line of $\angle$ APC
$\therefore \angle APO=\angle CPO={30}^{\circ }OA\perp AP$
Also, tangents at any point of a circle is perpendicular to the radius through the point of contact.
In right angled $\Delta OAP,$ we have

$\tan {30}^{\circ }=\frac{OA}{AP}=\frac{3}{AP}$

$\Rightarrow \frac{1}{\sqrt{3}}=\frac{3}{AP}$

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

$AP=CP=3\sqrt{3}cm$ [Tangents drawn from an external point are equal]

Hence, the length of each tangent is $3\sqrt{3}cm.$