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 each other. Find the length of the string of each phone.

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Solution

The positions of Ankur, Syed and David are represented as $A,B&C$ respectively.As they are sitting at equal distances, the triangle is equilateral$AD\perp BC$ is drawn. $AD$is the median of $\Delta ABC$and it passes through the centre $O$.$O$is the centroid of the $\Delta ABC$. $OA$is the radius of the triangle.$OA=\frac{2}{3}AD$Let the side of a triangle a metres then$BD=\frac{a}{2}$On applying Pythagoras theorem in $\Delta ABD$,$=A{B}^{2}=B{D}^{2}+A{D}^{2}\phantom{\rule{0ex}{0ex}}⇒A{D}^{2}=A{B}^{2}-B{D}^{2}\phantom{\rule{0ex}{0ex}}⇒A{D}^{2}={a}^{2}-{\left(\frac{a}{2}\right)}^{2}\phantom{\rule{0ex}{0ex}}⇒A{D}^{2}=\frac{3{a}^{2}}{4}\phantom{\rule{0ex}{0ex}}⇒AD=\frac{\sqrt{3}}{2}\phantom{\rule{0ex}{0ex}}Therefore,OA=\frac{2}{3}AD\phantom{\rule{0ex}{0ex}}20m=\frac{2}{3}×\frac{\sqrt{3}a}{2}\phantom{\rule{0ex}{0ex}}a=20\surd 3m$Hence , the length of the string of the toy is $20\sqrt{3}m$

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