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.

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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