Question

A circular park of radius 20m is situated in a calony. Three boys Ankur, Brijesh and chitprag are sitting at equal distances on its boundary, each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.​

Answer 1

$\bf\underline{\underline{\pink{Question:-}}}$

★A circular park of radius 20m is situated in a calony. Three boys Ankur, Brijesh and chitprag are sitting at equal distances on its boundary, each having a toy telephone in his hands to talk to each other. Find the length of the string of each phone.

$\bf\underline{\underline{\green{Solution:-}}}$

Let A, B and C be the position of snkur, Brijesh and Chitprag respectively.

Now, arc AB = arc BC = arc CA

==> chord AB = chord BC = chord CA

==>∆ABC is an equilateral triangle.

Let AD, BE and CFcbe the medians of the ∆ABC and let G be it's centeroid.

Since,in an equilateral triangle, the centeroid coincides with its circumcentre, we have

GA = GB = GC = 20m

Since the centeroid of a triangle devides a median in the ratio 2:1, we have

GA\GD = 2\1 => 20m\GD = 2\1 => GD = 10m

In right ∆BDG, we have

BG² = BD² + GD²

==> (20m)² = BD² + (10m)²

==> 400m² = BD² + 100m²

==> BD² = (400 – 100)m²

==> BD² = 300m²

==> BD = √300 m

==> BD = 10√3 m

∴ BC = 2 × BD = (2 × 10√3)m = 20√3 m.

Hence, the length of each telephone string is 20√3 m.