Question

A circular park of radius $40m$ is situated in a colony. Three boys Ankur, Amir, and Anand are sitting at an 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

The radius of circular park $=40m$

Ankur, Amir and Anand are sitting at equal distance to each other

By joining them, an equilateral triangle ABC is formed produce BO to L which is the perpendicular bisector of AC (which means BL is the median and altitude)

Medians of equilateral triangle pass through the circumcenter (O) of the equilateral triangle ABC. We also know that medians intersect each other at the $2:1$

i.e $\frac{OB}{OL}=\frac{2}{1}$

$OB=2OL$

$40=2\times OL$ [$OB$ is radius, and $OB=40cm$]

$\therefore OL=20cm$

$\therefore BL=OB+OL=40+20=60m$

($\because$ O is centroid of $\Delta ABC$ also )

Let ${}^{\prime }{a}^{\prime }$ be the side of $\Delta ABC$, BL is the also altitude of equilateral triangle ABC.

$\therefore \frac{\sqrt{3}}{2}a=60$ [$\because$ Altitude of equilateral triangle $=\frac{\sqrt{3}}{2}\times side$]

$\Rightarrow a=\frac{60\times 2}{\sqrt{3}}=\frac{120\times \sqrt{3}}{\sqrt{3}\times \sqrt{3}}$

$a=\frac{120\times \sqrt{3}}{3}=40\sqrt{3}m$

so, $a=AB=BC=CA=40\sqrt{3}m$

Hence the distance between each other $=40\sqrt{3}$ m