Line Segments that Subtends Equal Angles at Two Other Points
Question 3 In...
Question
Question 3 In the given figure, ∠PQR=100∘, where P, Q and R are points on a circle with centre O. Find ∠OPR.
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Solution
Consider PR as a chord of the circle. Take any point S on the major arc of the circle. The quadrilateral PQRS formed is a cyclic quadrilateral. ∠PQR+∠PSR=180∘(Opposite angles of a cyclic quadrilateral are supplementary) ∴∠PSR=180∘−100∘=80∘
We know that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the remaining part of the circle. ⇒∠POR=2∠PSR=2(80∘)=160∘InΔPOR, OP = OR (Radii of the same circle) ∴∠OPR=∠ORP(Angles opposite to equal sides of a triangle) ∠OPR+∠ORP+∠POR=180∘(Angle sum property of a triangle) 2∠OPR+160∘=180∘ 2∠OPR=180∘−160∘=20∘ ⇒∠OPR=10∘