Question

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.​

Answer 1

→ 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

$\implies$ ∆ ASD is an equilateral triangle.

Draw AB perpendicular to SD

Let AS be '2a' units.

so,

SD = DA = '2a' units

$\implies$ 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

$\therefore$ The length of the string of each phone will be 20√3 m .

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