Q. 4

In figure 2, PQ is a chord of a circle with center O and PT is a tangent. If QPT = 60°, find PRQ.


Answer :

Given : PQ is a chord of a circle with center O and PT is a tangent. Also QPT = 60°


To find : PRQ


Now OPT = 90° as OPPT


[As the tangent drawn at any point on a circle is perpendicular to the radius through the point of contact]


Given QPT = 60°


OPT - OPQ = 60°


90° - OPQ = 60°


OPQ = 30°


Now as OP = OQ [ radii of same circle]


OQP = QPQ = 30° [angle opposite to equal sides are equal]


In triangle OPQ


OPQ + POQ + OQP = 180° [Angle sum property of a triangle]


30 + POQ + 30 = 180


POQ = 120°


The angle subtended by an arc at the center is double the angle subtended by it at any point on the remaining part of the circle.


So,


Reflex POQ = 360 - 120 = 240°


Now PQ is an arc and we know


Reflex POQ = 2PRQ


PRQ = 240/2 = 120°


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