Ex 9.2, 6
A circular park of radius 20 m 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.
Given: Circular park with radius 20m
Let Ankur, Syed and David
be donated by points A,S & D resp.
Given all three are sitting at equal distances,
i.e., AS = SD = AD
To find: Length of AS = SD = AD
Explanation: Let AS = SD = AD = 2x
In ASD, all sides are equal,
∴ ASD is an equilateral triangle
We draw OP ⊥ SD
So, SP = DP = 1/2 SD
⇒ SP = DP = 2𝑥/2 = x
Join OS & AO
In Δ OPS
By Pythagoras theorem
OS2 = OP2 + PS2
(20)2 = OP2 + x2
400 = OP2 + x2
400 – x2 = OP2
OP2 = 400 – x2
OP = √("400 – x2" )
Now,
AP = AO + OP
√("3" )x = 20 + √("400 – x2" )
√("3" )x – 20 = √("400 – x2" )
√("400 – x2" ) = √("3" )x – 20
Squaring both sides
(√("400 – x2" ))^2= (√("3" )x – 20)2
400 – x2 = (√3x)2 + (20)2 – 2 ×(√3x) × (20)
"400 – x2 =" 3x2 + 400 – 40√3x
"400 – 400 + 40" √3 "x =" 3x2 + x2
"40" √3 "x =" 4x2
4x2 = 40√3x
"x2" /"x" = 40/4 √3
x = 10√3 m
AS = SD = AD = 2x = 2 × 10√3 = 20√3 m
∴ Length of string of each phone is 20√3 m

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.

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