Question

A circular park of radius 20 m 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 each other. Find the length of the string of each phone.

Answer 1

It is given that AS = SD = DA
Therefore, $\Delta ASD$ is an equilateral triangle.
The radius, OA = 20 m
The medians of the equilateral triangle pass through the circumcentre (O) of the equilateral triangle ASD.
We also know that medians intersect each other in the ratio 2:1.

Let AB be the median of the equilateral triangle ASD, so we can write
$\Rightarrow \frac{OA}{OB}=\frac{2}{1}$
$\Rightarrow \frac{20m}{OB}=\frac{2}{1}$
$\Rightarrow OB=\left(\frac{20}{2}\right)m=10m$
$\therefore AB=OA+OB=(20+10)m=30m$

In $\Delta ABD$,
$A{D}^2=A{B}^2+B{D}^2$[using pythagoras theorem]
$A{D}^2=(30{)}^2+{\left(\frac{DS}{2}\right)}^2$
We have AS = SD = DA,
$\therefore$ $A{D}^2=(30{)}^2+{\left(\frac{AD}{2}\right)}^2$
$A{D}^2=900+\frac{1}{4}A{D}^2$
$\frac{3}{4}A{D}^2=900$
$A{D}^2=1200$
$AD=20\sqrt{3}$
Therefore, the length of the string of each phone will be $20\sqrt{3}m$.