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In the given figure, PQ is a chord of a circle with center 0 and PT is a tangent. If ∠QPT = 60°, find ∠P
Given : , PQ is a chord of a circle with center 0 and PT is a tangent and ∠QPT = 60°.
To Find : ∠PRQ
∠OPT = 90°
∠OPQ + ∠QPT = 90°
∠OPQ + 60° = 90°
∠OPQ = 30° … [1]
Also.
OP = OQ [radii of same circle]
∠OQP = ∠OPQ [Angles opposite to equal sides are equal]
From [1], ∠OQP = ∠OQP = 30°
In △OPQ , By angle sum property
∠OQP + ∠OPQ + ∠POQ = 180°
30° + 30° + ∠POQ = 180°
∠POQ = 120°
As we know, 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, we have
2∠PRQ = reflex ∠POQ
2∠PRQ = 360° - ∠POQ
2∠PRQ = 360° - 120° = 240°
∠PRQ = 120°