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 in his hands to talk to each other. Find the length of the string of each phone.

Answer 1

Let Ankur be represented as $A$, Syed as $S$ and David as $D$.

The boys are sitting at an equal distance.

Hence, $△ASD$ is an equilateral triangle.

Let the radius of the circular park be $r$ meters.

$\therefore OS=r=20$m.

Let the length of each side of $△ASD$ be $x$ meters.

Draw $AB\perp SD$

$\therefore SB=BD=\frac{1}{2}SD=\frac{x}{2}$ m

In $△ABS,\angle B={90}^o$

By Pythagoras theorem,

$A{S}^2=A{B}^2+B{S}^2$

$\therefore A{B}^2=A{S}^2-B{S}^2=x^2-{\left(\frac{x}{2}\right)}^2$$=\frac{3x^2}{4}$

$\therefore AB=\frac{\sqrt{3}x}{2}$ m

Now, $AB=AO+OB$

$OB=AB-AO$

$OB=(\frac{\sqrt{3}x}{2}-20)$ m

In $△OBS$,

$O{S}^2=O{B}^2+S{B}^2$

${20}^2=$${(\frac{\sqrt{3}x}{2}-20)}^2+{\left(\frac{x}{2}\right)}^2$

$400=\frac{3}{4}{x}^2+400-2\left(20\right)\left(\frac{\sqrt{3}x}{2}\right)+\frac{x^2}{4}$

$0=x^2-20\sqrt{3}x$

$\therefore x=20\sqrt{3}$m

The length of the string of each phone is $20\sqrt{3}$m.