Q: 6     A circular park of radius \small 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.

Answers (1)
S seema garhwal

Given: In the figure, A,S,D are positions Ankur, Syed and David respectively.

So, AS = SD = AD 

Radius of circular park = 20 m

  so, AO=SO=DO=20 m

Construction : AP\perp SD

Proof :


Let AS = SD = AD = 2x cm

In \triangleASD,

  AS = AD   and     AP\perp SD

So,  SP = PD = x cm 

In \triangleOPD, by pythagoras,


\Rightarrow OP^2=20^2-x^2=400-x^2

\Rightarrow OP=\sqrt{400-x^2}

In \triangleAPD, by pythagoras,


\Rightarrow (AO+OP)^2+x^2=(2x)^2

\Rightarrow (20+\sqrt{400-x^2})^2+x^2=4x^2

\Rightarrow 400+400-x^2+40\sqrt{400-x^2}+x^2=4x^2

\Rightarrow 800+40\sqrt{400-x^2}=4x^2

\Rightarrow 200+10\sqrt{400-x^2}=x^2

\Rightarrow 10\sqrt{400-x^2}=x^2-200

Squaring both sides,

\Rightarrow 100(400-x^2)=(x^2-200)^2

\Rightarrow 40000-100x^2=x^4-40000-400x^2

\Rightarrow x^4-300x^2=0

\Rightarrow x^2(x^2-300)=0

\Rightarrow x^2=300

\Rightarrow x=10\sqrt{3}

Hence, length of string of each phone= 2x=20\sqrt{3}m