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.

The figure is shown as:



 


It is given that AS = SD = DA


⇒ ΔASD is an equilateral triangle


Now, OA (radius) = 20 m


And, the medians of an equilateral triangle pass through the circumcentre (O) of the equilateral triangle ASD.


 


Now, We also know that medians intersect each other in the ratio 2: 1.


 


As, AB is the median of equilateral triangle ASD,


∴ OA/OB = 2/1


⇒ 20m/OB = 2/1


⇒ OB = (20/2)m


⇒ OB = 10 m


 


Now, AB = OA + OB


⇒  AB = (20 + 10) m


⇒  AB = 30 m


Now, In ΔABD,


 


AD2 = AB2 + BD2


⇒  AD2 = (30)2 + (SD/2)2


⇒  AD2 = (30)2 + (AD/2)2


(Because AS = SD = DA in the equilateral triangle)


⇒ AD2 = 900 + (1/4)AD2


⇒  AD2 -  (1/4)AD2 = 900


⇒  (3/4)AD2  = 900


⇒  AD2 = 1200


⇒  AD = 20√3 m


Therefore, the length of the string of each phone will be 20√3 m


 

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