Question

A circular park of radius 40 m is situated in a colony. Three boys Ankur, Amit and hands to talk to each other. Find the length of the string of each phone.

Answer 1

From the given data, we see that the given situation is equivalent to an equilateral triangle circumscribed by a circle.

Let the positions of the three boys Ankur, Amit and Anand be denoted by the points ‘A’,’B’ and ‘C’. Let ‘O’ be the centre of the circle, ‘a’ is the sides of the equilateral triangle and ‘R’ is its circumradius.

Now, in an equilateral triangle with side ‘a’, the height, ‘h’ of the equilateral triangle would be,

AB = BC = CA
Therefore, $∆ABC$is an equilateral triangle.
OA = 40 m
Medians of equilateral triangle pass through the circumcentre (O) of the equilateral triangle ABC. We know that medians intersect each other in the ratio 2 : 1. As AD is the median of equilateral triangle ABC, we can write

$$\begin{gathered} \frac{OA}{OD}=\frac{2}{1} \\ \Rightarrow \frac{40}{OD}=\frac{2}{1} \\ \Rightarrow OD=20\mathrm{m} \\ AD=AO+OD=\left(40+20\right)\mathrm{m}=60\mathrm{m} \\ \mathrm{In}∆ADC, \\ A{C}^2=A{D}^2+D{C}^2 \\ \Rightarrow A{C}^2={\left(60\right)}^2+\frac{A{C}^2}{4}\left[\because AC=BC,\mathit{}DC=\frac{1}{2}BC\Rightarrow DC=\frac{1}{2}AC\right] \\ \Rightarrow \frac{3A{C}^2}{4}=3600 \\ \Rightarrow A{C}^2=4800 \\ \Rightarrow AC=40\sqrt{3}\mathrm{m} \end{gathered}$$
Hence the length of the string of each phone is