Question

In Figure, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.​

Answer 1

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Answer 2

Answer:

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Solution:

Take any point A on the circumcircle of the circle.

Join AP and AR.

∵ APQR is a cyclic quadrilateral.

∴ ∠PAR + ∠PQR = 180° [sum of opposite angles of a cyclic quad. is 180°]

∠PAR + 100° = 180°

⇒ Since ∠POR and ∠PAR are the angles subtended by an arc PR at the centre of the circle and circumcircle of the circle.

∠POR = 2∠PAR = 2 x 80° = 160°

∴ In APOR, we have OP = OR [radii of same circle]

∠OPR = ∠ORP [angles opposite to equal sides]

Now, ∠POR + ∠OPR + ∠ORP = 180°

⇒ 160° + ∠OPR + ∠OPR = 180°

⇒ 2∠OPR = 20°

⇒ ∠OPR = 10°

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