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.

Question

Philoid · Accepted Answer

The figure is shown as:

It is given that AS = SD = DA

⇒ ΔASD is an equilateral triangle

Now, OA (radius) = 20 m

And, the medians of an equilateral triangle pass through the circumcentre (O) of the equilateral triangle ASD.

Now, We also know that medians intersect each other in the ratio 2: 1.

As, AB is the median of equilateral triangle ASD,

∴ OA/OB = 2/1

⇒ 20m/OB = 2/1

⇒ OB = (20/2)m

⇒ OB = 10 m

Now, AB = OA + OB

⇒ AB = (20 + 10) m

⇒ AB = 30 m

Now, In ΔABD,

AD² = AB² + BD²

⇒ AD² = (30)² + (SD/2)²

⇒ AD² = (30)² + (AD/2)²

(Because AS = SD = DA in the equilateral triangle)

⇒ AD² = 900 + (1/4)AD²

⇒ AD² - (1/4)AD² = 900

⇒ (3/4)AD² = 900

⇒ AD² = 1200

⇒ AD = 20√3 m

Therefore, the length of the string of each phone will be 20√3 m