Question

Question 9
If two tangents inclined at an angle ${60}^{\circ }$ are drawn to a circle of radius 3 cm, then the length of each tangent is
(A) $\frac{3}{2}$$\sqrt{3}$cm
(B) 6 cm
(C) 3 cm
(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 ${60}^{\circ }$


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.
$Inrightangled\Delta OAP,tan{30}^{\circ }=\frac{OA}{AP}=\frac{3}{AP}\Rightarrow \frac{1}{\sqrt{3}}=\frac{3}{AP}\Rightarrow AP=3\sqrt{3}cm$
Hence, the length of each tangent is $3\sqrt{3}cm.$