Q

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.

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.

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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 $\triangle$ASD,

AS = AD   and     AP$\perp$ SD

So,  SP = PD = x cm

In $\triangle$OPD, by pythagoras,

$OP^2=OD^2-PD^2$

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

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

In $\triangle$APD, by pythagoras,

$AP^2=AD^2-PD^2$

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

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

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

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

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

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

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

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