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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=180100=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=2PSR=2(80)=160In ΔPOR,
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)
2OPR+160=180
2OPR=180160=20
OPR=10

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