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A circular park of radius 20m is situ...

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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