Angle Subtended by an Arc of a Circle at the Centre
In the figure...
Question
In the figure below, O is the centre of the circle and ∠QPR=x∘,∠ORQ=y∘. Which statement is true about x∘ and y∘?
A
x∘+y∘=120∘
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B
x∘+y∘=180∘
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C
x∘+y∘=90∘
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D
x∘+y∘=150∘
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Solution
The correct option is Cx∘+y∘=90∘ Angle subtended by an arc of a circle at the centre of the circle is double the angle subtended by it at any other point on the circle. So, ∠QOR=2∠QPR=2x. Note that OQ=OR (∵ radius of the same circle) ⟹∠OQR=∠ORQ (∵ angles opposite to equal sides are equal) In ΔOQR, 2x∘+y∘+y∘=180∘ i.e., 2x∘+2y∘=180∘ ∴x∘+y∘=90∘