Two circles are concentric. A chord c units long cuts across the larger, tangent to the smaller. What is the area of the shaded region in terms of c?
Solution: [From MT Nov 1991]
Let the radii of the smaller and larger circles be r and R respectively, so the area we seek is given by ℼR2 – ℼr2. Now, let A be the center of both circles, let point B be the point of tangency of the chord, and D be the point of intersection with the larger circle, as indicated. Now, because triangle ABD is right, we have r2 + (c/2)2 = R2, so that R2 – r2 = (c/2)2. The shaded area is then equal to ℼc2/4.
