If a tangent PQ is drawn at point P to a circle such that ∠OPQ = 90°, and a chord PR is drawn such that ∠RPQ = 30°, what is the angle subtended by the chord PR at the centre O?

Solution

Solution: Since OP is perpendicular to the tangent PQ, \(\angle OPR=90^\circ-30^\circ=60^\circ\). In triangle OPR, OP and OR are radii, so OP = OR and the base angles are equal. Thus, \(\angle ORP=60^\circ\), and \(\angle POR=180^\circ-60^\circ-60^\circ=60^\circ\). Trick: Use the perpendicular radius-tangent relationship and the isosceles triangle formed by two radii.