Question

Two congruent circles with centers $A$ and $B$ intersect each other at $C$ and $D$. Circle with center $A$ passes through $B$. Point $P$ lies on the circle with center $B$. $A, B, P$ are not collinear. Find m$\angle APC$.[2013]

Answer 1

30

Answer 2

Solution Explanation

1. Since the circles are congruent and the circle with center A passes through B, we have AB equal to the radius of both circles. Hence, triangle ABC (with C as one intersection point) is equilateral (AB = BC = CA).

2. In an equilateral triangle, each central angle is 60°. Therefore, the intercepted arc AC on the circle centered at B is 60°.

3. Now, by the Inscribed Angle Theorem, any angle subtended by arc AC at a point P on the circle (other than the ends of the arc) is half the central angle. Thus, ∠APC = ½ × 60° = 30°.