2004 AMC 10B Problem 12

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

The annulus area is

\pi b^2 - \pi c^2

Since

\overline{XZ}

is tangent at

Z,

triangle

OZX

is right-angled, giving

b^2 - c^2 = a^2

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.

Problem 12 in other years