Fig. 10.25
6. 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.
Answers
→ Let A,S and D represents the position of Ankur, Syed and David respectively.
→ Let O be the centre of the circular path.
AS = SD = DA
∆ ASD is an equilateral triangle.
Draw AB perpendicular to SD
Let AS be '2a' units.
so,
SD = DA = '2a' units
SB = BD = '2a'/2 = 'a' m
Now,
OA = 20 m [As radius of circular park is 20m]
In right ∆ABS
AS² = AB² + BS² [By using Pythagoras theorem]
→ 4a² = AB² + a²
→ AB² = 4² - a²
→ AB² = √3a² m
→ AB² = √3a m
OB = AB - OA
OB = (√3a - 20) m
In right ∆ OBS
OS² = OB² + BS² [By using Pythagoras theorem]
→ (20)² = (√3a - 20)² + a²;
→ 400 = (√3a)² + (20)² - 40√3a + a²
→ 400 = 3a² + 400 - 40√3a + a²
→ 4a² + 400 - 40√3a - 400 = 0
→ 4a (a - 10√3) = 0 (as a ≠ 0)
→ a = 10√3
Now,
AS = SD = DA = '2a' units
⠀⠀⠀⠀⠀⠀ ⠀⠀= 2 × 10√3 m
⠀⠀⠀⠀⠀⠀ ⠀⠀= 20√3 m
The length of the string of each phone will be 20√3 m .
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