Question

Two tangents making an angle of ${120}^o$ with each other are drawn to a circle of radius $6cm$, find the length of each tangent.

Answer 1

We have,

Let the center of circle be $O$.

Let the tangents be $PA$ and $PB$ from point $P$ into the circle .

$OA=OB$=radius=6cm$.

Then,

In the triangle $OAP$ and $OBP$:

$\angle OAP=\angle OBP={90}^0$

$\begin{array}{c}\angle OPA=\angle OPB=\frac{{120}^0}{2}={60}^0\\ \angle POA=\angle POB=\frac{{120}^0}{2}={30}^0\end{array}$

$Tan60=OA/PA$

$PA=OA$ cot ${60}^0=\frac{6}{\sqrt{3}}=2\sqrt{3}$

Then,

The length of tangent $=2\sqrt{3}$cm.