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.

Solution

Given

The positions of Ankur, Syed and David are represented as A, B and C respectively.

As they are sitting at equal distances, the triangle is equilateral

AD ⊥ BC is drawn. 

AD is the median of ΔABC and it passes through the centre O.

O is the centroid of the ΔABC. OA is the radius of the triangle.

OA = 2/3 AD

Let the side of a triangle a metres then BD = a/2 m.

On applying Pythagoras theorem in ΔABD,

AB= BD2+AD2

⇒ AD= AB-BD2

⇒ AD= a-(a/2)2

⇒ AD= 3a2/4

⇒ AD = √3a/2

OA = 2/3 AD

20 m = 2/3 × √3a/2

a = 20√3 m

Answer

The length of the string of the toy is 20√3 m.

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