In circle geometry, AB is a diameter of a circle with centre O. CD is a chord such that chord CD is equal in length to the radius OC. Lines AC and BD are produced (extended) to meet at an external point P outside the circle. What is the measure of angle ∠APB?

Introduction / Context:This question tests circle theorems about chords, arcs, and the angle formed by two secants intersecting outside a circle. The key observation is that if a chord length equals the radius, it subtends a fixed central angle of 60°. Then, the external secant angle formula converts intercepted arcs into the required angle ∠APB.

Given Data / Assumptions:

• AB is a diameter, so the arc AB (a semicircle) measures 180°.
• CD is a chord with CD = radius OC.
• Secants through P intersect the circle at (A, C) and (B, D) respectively.

Concept / Approach:1) Chord-length and central angle: for a circle of radius r, a chord of length r satisfies: Chord = 2r*sin(theta/2). If chord = r, then 2r*sin(theta/2) = r, so sin(theta/2) = 1/2, giving theta/2 = 30° and theta = 60°. Thus arc CD = 60°. 2) External secant angle theorem: The angle formed by two secants from an external point equals half the difference of the intercepted arcs: ∠APB = (1/2) * (arc AB - arc CD).

Step-by-Step Solution:

Verification / Alternative check:A chord equal to radius always corresponds to a 60° central angle, so arc CD = 60° is fixed. With AB a diameter, arc AB is fixed at 180°. Therefore the external secant angle is uniquely determined and must be 60°.

Why Other Options Are Wrong:

Common Pitfalls:Confusing inscribed angle rules with external secant angle rules, or assuming chord = radius implies a 90° angle (it actually implies 60°). Also, forgetting that diameter corresponds to a semicircle arc of 180° is a frequent slip.

Final Answer:∠APB = 60°