# 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 to

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

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Let Ankur, Syed and David standing on the points P,Q and R.
Let PQ=QR=PR=x
Therefore, DeltaPQR is an equilateral triangle. Drawn altitudes PC, OD and RN form vertices to the sides of a triangle and intersect these altitude at the centre of a circle M.
As PQR is an equilateral, therefore these altitude bisects their sides.
In DeltaPQC
PQ^2=PC^2+QC^2 (by pythagorus theorem)
x^2=PC^2+((x)/(2))^2
rArr PC^2=x^(2)-(x^2)/(4)=(3x^2)/(4)" "(because QC=(1)/(2)QR=(x)/2)
rArrPC=sqrt(3x)/(2)
Now, MC=PC-PM=sqrt(3x)/(2)-20" "(becausePM="radius"=20 m)
In DeltaQCM,
QM^2=QC^2+MC^2
(20)^2=(x/2)^2+(sqrt(3x)/(2)-20)^2" "(because QM ="radius"=20 m)
rArr400=(x^2)/(4)+-20sqrt(3x)+400
rArr0=x^2-20sqrt(3x)
rArrx=20sqrt(3)m
HEnce, PQ=QR =PS= 20(sqrt3)m.

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