Question

An annulus is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c,$ with $b \gt c.$ Let $\overline{OX}$ be a radius of the larger circle, let $\overline{XZ}$ be tangent to the smaller circle at $Z,$ and let $\overline{OY}$ be the radius of the larger circle that contains $Z.$ Let $a = XZ,$ $d = YZ,$ and $e = XY.$ What is the area of the annulus?

$\pi a^2$

$\pi b^2$

$\pi c^2$

$\pi d^2$

$\pi e^2$

Solution:

The annulus is the difference of the two circular areas, $\pi b^2 - \pi c^2.$

Because $\overline{XZ}$ is tangent to the small circle at $Z,$ it is perpendicular to the radius $\overline{OZ}.$ In right triangle $OZX$ with $OX = b,$ $OZ = c,$ and $XZ = a,$ we get $b^2 - c^2 = a^2.$

Therefore the area of the annulus is $\pi a^2.$

Thus, the correct answer is A.

2004 AMC 10B Problem 12

Solution:

Problem 12 in Other Years