Question

A circular park of radius 40 m is situated in a colony. Three boys Ankur, Amit and Anand 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

Given that $AB=BC=CA$

So, $ABC$ is an equilateral triangle

$OA$ (radius)=$40m$

Median of equilateral triangle pass through the circum center $\left(0\right)$ of the equilateral triangle $ABC$

We also know that median intersect each other at the $2:1$

As $AD$ is the median of equilateral triangle $ABC$, we can write$:$

$\begin{array}{c}\frac{OA}{OD}=\frac{2}{7}\\ \Rightarrow \frac{40m}{OD}=\frac{2}{7}\\ \Rightarrow OD=20m\\ \therefore AD=OA+OD=\left(40+20\right)m\\ =60m\\ In\Delta ADC\\ Byu\sin gpythagorastheorem\\ A{C}^2=A{D}^2+D{C}^2\\ A{C}^2=A{D}^2+D{C}^2\\ A{C}^2={\left(60\right)}^2+{\left(\frac{AC}{2}\right)}^2\\ A{C}^2=3600+\frac{A{C}^2}{4}\\ \Rightarrow \frac{3}{4}A{C}^2=3600\\ \Rightarrow A{C}^2=4800\\ \Rightarrow AC=40\sqrt{3}m\\ So,lengthofstringofeachphonewillbe40\sqrt{3}m\end{array}$