Question

A circular park of radius $20m$ is situated in a colony. Three boys Ankur, Syed and David are sitting at equal distance on its boundary each having a toy telephone his hands to talk each other. Find the length of the string of each phone.

Answer 1

A is the position of Ankur.

B is the position of Syed.

C is the position of David.

In the equilateral triangle ABC, altitude, median, and angle bisector are all same.

$\Rightarrow OB$ bisects $\angle B$ [$\because$ OB is the angle bisector]

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

In $\Delta OBD,\cos {30}^{\circ }=\frac{BD}{OB}=\frac{\frac{a}{2}}{R}$ [$\because R=20m=$ radius of the circle]

$\Rightarrow a=2R\cos {30}^{\circ }$

$\Rightarrow a=2\times 20\times \frac{\sqrt{3}}{2}=20\sqrt{3}m$ [$\because \cos {30}^{\circ }=\frac{\sqrt{3}}{2}$ ]

Hence, the length of the string is $20\sqrt{3}m$