NCERT Solutions Class 9 Mathematics Solutions for Circles - Exercise 10.4 in Chapter 10 - Circles
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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 to each other. Find the length of the string of each phone.
First, draw a diagram according to the given statements. The diagram will look as follows.
Here the positions of Ankur, Syed, and David are represented as A, B, and C respectively. Since they are sitting at equal distances, the triangle ABC will form an equilateral triangle.
AD ⊥ BC is drawn. Now, AD is the median of ΔABC and it passes through the center O.
Also, O is the centroid of the ΔABC. OA is the radius of the triangle.
OA = 2/3 AD
Let the side of a triangle meters then BD = a/2 m.
Applying the Pythagoras theorem in ΔABD,
\begin{array}{l} 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}-(a / 2)^{2} \\ \Rightarrow A D^{2}=3 a^{2} / 4 \\ \Rightarrow A D=\sqrt{3} a / 2 \\ O A=2 / 3 A D \\ \Rightarrow 20 m=2 / 3 \times \sqrt{3} a / 2 \\ \Rightarrow a=20 \sqrt{3} m \end{array}
So, the length of the string of the toy is 20√3 m.
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