Question

\(P,Q,R\) are points on a circle with centre \(O\) as shown in the figure. If \(\angle PQR=100^\circ\), then \(\angle OPR=\)

• A. \(10^\circ\)
• B. \(20^\circ\)
• C. \(30^\circ\)
• D. \(15^\circ\)

Hint:

Always check whether the inscribed angle gives the major or minor arc before doubling.

Answer 1

The Correct Option is A

Angle subtended by chord \(PR\) at the circumference: \[ \angle PQR=100^\circ \] The angle subtended by the same chord at the centre is twice the angle at the circumference. So major angle: \[ \angle POR=2\times100^\circ=200^\circ \] Thus minor central angle: \[ 360^\circ-200^\circ=160^\circ \] In triangle \(OPR\), \[ OP=OR \] (radius of circle) So base angles are equal: \[ \angle OPR=\angle ORP \] Using angle sum: \[ \angle OPR+\angle ORP+160^\circ=180^\circ \] \[ 2\angle OPR=20^\circ \] \[ \angle OPR=10^\circ \] Thus, \[ \boxed{10^\circ} \]