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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.
Asked by | 04 Mar, 2013, 06:37: PM
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)
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
Answered by | 05 Mar, 2013, 12:12: AM

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