Question

In the following, PQ is a chord of a circle and PT is the tangent at P such that ∠QPT = 60°. Then, find ∠PRQ.

Answer 1

Given: PQ is a chord of the circle, PT is the tangent at P, and ∠QPT = 60°.

Step-wise calculation:

1. Mark a point M on the circle in the alternate segment (i.e., on the arc opposite the tangent at P). By the alternate-segment theorem, the angle between the tangent and the chord equals the angle in the opposite arc, so ∠PMQ = ∠QPT = 60°.

2. Points P, M, Q, R lie on the same circle, so PMQR is a cyclic quadrilateral. Opposite angles of a cyclic quadrilateral sum to 180°, hence ∠PMQ + ∠PRQ = 180°.

3. Substitute ∠PMQ = 60° to get ∠PRQ = 180° – 60° = 120°.