# Two tangents making an angle of 120° with each other are drawn

Question:

Two tangents making an angle of 120° with each other are drawn to a circle of radius 6 cm, then the length of each tangent is equal to

(a) $\sqrt{3} \mathrm{~cm}$

(b) $6 \sqrt{3} \mathrm{~cm}$

(c) $\sqrt{2} \mathrm{~cm}$

(d) $2 \sqrt{3} \mathrm{~cm}$

Solution:

We are given two tangents to a circle making an angle of 120° with each other. The radius of circle is 6 cm

We have to find the length of each tangent.

Let be the center of the given circle

Let AB and AC be the two tangents to the given circle drawn from point A

Therefore

Now OB and OC represent the radii of the circle

Therefore

[Since Radius of a circle is perpendicular to tangent]

In $\triangle A B O$ and $\triangle A C O$

$O B=O C$ [Radii of same circle]

$O A=O A \quad[$ Common Side $]$

$A C=A B$ [Tangent segments to a circle from an external point are equal]

Therefore $A B O$ and $\triangle A C O$ are congruent by SSS rule

Hence $\angle B A O=\angle C A O$ [Corresponding angles of congruent triangles]

Now $\angle B A O+\angle C A O=120^{\circ}$

$2 \angle B A O=120^{\circ}$

$\angle B A O=60^{\circ}$

In right $\triangle A B O$

$\frac{O B}{A B}=\tan (\angle B A O)$

$\frac{O A}{A B}=\tan 60^{\circ}$

$\frac{O A}{A B}=\sqrt{3}$

$A B=\frac{O A}{\sqrt{3}}$

$A B=\frac{6}{\sqrt{3}}$

$=\frac{6}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}}$

$=2 \sqrt{3}$

Hence option (d) is correct.