Mon March 04, 2013 By:

on a circle's boundary 3 boys are standing equidistant from each other having a toy phone in hands to talk to each other.find the length of the phone's string if the radius of the circle is 20m.

Expert Reply
Tue March 05, 2013
Let A, B and C represent the 3 boys standing equidistant from one another on a circle's boundary. 
ABC form an equilateral triangle and the center of the circle O represents the circumcenter. 
Draw a perpendicular OM from point O to side AB of the triangle. 
Now, since a perpendicular from the center to a chord of a circle (i.e. AB here) bisects it. 
So, MB = MA = 1/2 AB
Now, in triangle MOB, angle OMB = 90 and hence, it is a right angled triangle.
Furthermore,in triangle AOB and triangle COB 
OB = BO (common)
AB = BC (sides of equilateral triangle)
AO = OC (radius)
triangle AOB is congruent to triangle COB  (By SSS)
hence, angle(OBA) = angle(OBC) (By CPCT)
Now, all angles of an equilateral triangle are equal to 60 degree. 
So, angle(OBA)+ angle(OBC) = 60
And since, they are equal, we get angle(OBA) = 30 degree. 
Now in triangle MOB 
using trignometry
MB = cos30 * OB
MB = sqrt(3)/2 * 20 = 10*sqrt(3)
Since, AB = 2*MB 
So, AB = 20sqrt(3) cm
So,the length of the phone string = 20sqrt(3) cm
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