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.

Answer 1

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 $\Delta ABD$,

$$\begin{gathered} =A{B}^2=B{D}^2+A{D}^2 \\ \Rightarrow A{D}^2=A{B}^2-B{D}^2 \\ \Rightarrow A{D}^2=a^2-{\left(\frac{a}{2}\right)}^2 \\ \Rightarrow A{D}^2=\frac{3a^2}{4} \\ \Rightarrow AD=\frac{\sqrt{3}}{2} \\ Therefore,OA=\frac{2}{3}AD \\ 20m=\frac{2}{3}\times \frac{\sqrt{3}a}{2} \\ a=20\surd 3m \end{gathered}$$

Hence , the length of the string of the toy is $20\sqrt{3}m$