jmc

Question

The figure shows two concentric circles. If the length of chord AB is 80 units and chord AB is tangent to the smaller circle, what is the area of the shaded region? Express your answer in terms of

π

.

Accepted Answer

Call the point of tangency between the two circles P and the center O. FIGURE OPAB, so OP bisects AB. This means AP=40. By the Pythagorean Theorem, AP^2=1600=AO^2-OP^2. The area of the shaded region is AO^2-OP^2=(AO^2-OP^2)=1600 square units.