Question

If two tangents inclined at an angle of ${60}^{\circ }$ are drawn to a circle of radius $4$ cm, then the length of each tangent is equal to:

• A. $2\sqrt{3}$ cm
• B. $8$ cm
• C. $4$ cm
• D. $4\sqrt{3}$ cm

Answer 1

Answer: (D)

$4\sqrt{3}$ cm

Consider a circle of radius $4$ cm and center $O$. Tangents are drawn from point $P$ to the circle at $A$ and $B$.
A line form the center bisects the angle between the tangents.
Hence, $\angle APO={30}^{\circ }$
Also, $\angle OAP=90$ ....(Angle between tangent and the radius)
Thus, $\tan \angle APO=\frac{OA}{AP}$
$\Rightarrow \tan 30=\frac{4}{AP}$
$\Rightarrow AP=4\sqrt{3}$ cm