Question

If two tangents inclined at an angle${60}^{\circ }$are drawn to a circle of the radius $3cm$ then the length of each tangent is

Answer 1

Solve for the length of each tangent.

Given:

$$\begin{gathered} AO=OB=3cm \\ \angle APO=\angle OPB \end{gathered}$$

To find:

Relation between lengths of tangent

Solve:

In the given figure,

$AO=OB$(Given)

$\angle APO=\angle OPB$(Given)

$OP=OP$(Common)

$△AOP\cong △BOP$(By SSA rule)

Since,

$\angle APO=\angle BPO$

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

In$△AOP,OA\perp AP$

Therefore,

$$\begin{gathered} \tan {30}^{\circ }=\frac{OA}{AP} \\ \Rightarrow \frac{1}{\sqrt{3}}=\frac{3}{AP} \\ \Rightarrow AP=3\sqrt{3}cm \end{gathered}$$

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