Question

In Fig. 9.24, ∠PQR = 100°, where P, Q and R are points on a circle with centre O. Find ∠OPR.

Answer 1

Consider PR as a chord of the circle. 

Take any point S on the major arc of the circle. 

PQRS is a cyclic quadrilateral. 

∠PQR +∠ PSR = 180° (Opposite angles of a cyclic quadrilateral) 

∠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° 

In ∆POR, 

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) 

∠OPR + 160° = 180° 2 

∠OPR = 180° − 160° = 20º 2 

∠OPR = 10°