Question

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 please ☺️

Answer 1

Let positions of three boys Ankur, Syed and David be denoted by points A,B and C.

Three points are at equal distances.

Therefore, AB = BC = AC = a m (say)

Equal sides of equilateral triangles are equal as equal chords and perpendicular distances of equal chords of a circle are equidistant from the centre.

Therefore, OD = OE = OF = x (say)

Join OA, OB and OC.

Now, we have three congruent triangles,

∆OAB, ∆OBC and ∆AOC

Therefore, area (∆AOB) = area (∆BOC)
= area (∆AOC) -----(1)

Now, area of equilateral triangle ABC of side 'a'

= ar(∆AOB) + ar(∆BOC) + ar(∆AOC) ---(2)

=> Area (∆ABC) = 3Area (∆BOC) [using (1) and (2)]

$= > \: \frac{ \sqrt{3} }{4} {a}^{2} = ( \frac{1}{2} BC \times OE) \\ \\ \frac{ \sqrt{3} }{4} {a}^{2} = 3( \frac{1}{2} \times a \times x) \\ \\ \frac{ {a}^{2} }{a} = 3 \times \frac{1}{2} \times \frac{4}{ \sqrt{3} } \times x \\ \\ = > \: a = 2 \sqrt{3} x \: \: \: \: ...(3) \\ \\ also \: \: OE \: is \: perpendicular \: at \: BC \\ \\ therefore, \: \: BE = EC \: = \frac{1}{2} BC \\ (because \: perpendicular \: drawn \: from \: the \: centre \: bisects \: the \: chord)$

$= > \: BE \: = EC = \frac{1}{2} a \\ \\ = > \: BE \: = EC \: = \frac{1}{2} (2 \sqrt{3} x) \: \: \: (using \: (3)) \\ \\ = > \: BE \: = EC \: = \sqrt{3} x. \\ \\ now \: in \: rt. \: triangle \: BEO \\ \\ {OE}^{2} + {BE}^{2} = {OB}^{2} \: \: \: (pythagoras \: theorem) \\ \\ = > {x}^{2} + ( \sqrt{3} x {)}^{2} = {20}^{2} \\ \\ = > {x}^{2} + {3x}^{2} = 400 \\ \\ = > {4x}^{2} = 400 \\ \\ = > {x}^{2} = \frac{400}{4} \\ \\ {x}^{2} = 100 \\ \\ = > x = \sqrt{100} \\ \\ x = 10 \: m \: \: \: \: ....(4)$

$now \: from \: (3) \\ \\ a = 2 \sqrt{3} x \\ \\ = > \: a = 2 \sqrt{3} \times 10m \: \: \: (using \: (4)) \\ \\ = > \: a = 20 \sqrt{3} m \\ \\ hence \: the \: distance \: between \: any \: two \: boys \: is \: 20 \sqrt{3} m.$

Answer 2

Hey mate

Your answer is in attachment

Answer :-
$20 \sqrt{3}$
Hope it helps uh