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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 to each other. Find the length of the string of each phone.

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Solution

## Let Ankur be represented as A, Syed as S and David as D.The boys are sitting at an equal distance.Hence, △ASD is an equilateral triangle.Let the radius of the circular park be r meters.∴OS=r=20m.Let the length of each side of △ASD be x meters.Draw AB⊥SD∴SB=BD=12SD=x2 mIn △ABS,∠B=90oBy Pythagoras theorem,AS2=AB2+BS2∴AB2=AS2−BS2=x2−(x2)2=3x24 ∴AB=√3x2 mNow, AB=AO+OBOB=AB−AOOB=(√3x2−20) mIn △OBS,OS2=OB2+SB2202=(√3x2−20)2+(x2)2400=34x2+400−2(20)(√3x2)+x24 0=x2−20√3x∴x=20√3mThe length of the string of each phone is 20√3m.

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