Question

If two tangents inclined at an angle of 60⁰ are drawn to a circle of radius 3 cm , then length of each tangent is equal to
(a) $\frac{3\sqrt{3}}{2}$ (b) 6 cm (c) 3 cm (d) $3\sqrt{3}$ cm

Answer 1

Let PA and PB be two tangents to a circle with centre O and radius 3 cm.

We are given ∠APB = 60°

We know that two tangents drawn to a circle from an external point are equally inclined to the segment joining the centre to the point.

∴ ∠APO = ∠BPO = $\frac{1}{2}$ × ∠APB = $\frac{1}{2}$ × 60° = 30°

Also, OA ⊥ AP and OB ⊥ BP (radius ⊥ tangent at point of contact)

In right ΔOAP,
$$\begin{gathered} \tan 30°=\frac{3}{PA} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{3}{PA} \\ \Rightarrow PA=3\sqrt{3} \end{gathered}$$

Therefore, PA = PB = $3\sqrt{3}\mathrm{cm}$
Hence, the correct answer is option (d).