MathDB

An annulus is the region between two concentric circles. The concentric circles in the ﬁgure have radii

b

and

c

, with

b > 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 \equal{} XZ, d \equal{} YZ, and e \equal{} XY. What is the area of the annulus?

<span class='latex-bold'>(A)</span>\ \pi a^2 \qquad <span class='latex-bold'>(B)</span>\ \pi b^2 \qquad <span class='latex-bold'>(C)</span>\ \pi c^2 \qquad <span class='latex-bold'>(D)</span>\ \pi d^2 \qquad <span class='latex-bold'>(E)</span>\ \pi e^2

[asy]unitsize(1.4cm); defaultpen(linewidth(.8pt)); dotfactor=3;
real r1=1.0, r2=1.8; pair O=(0,0), Z=r1*dir(90), Y=r2*dir(90); pair X=intersectionpoints(Z--(Z.x+100,Z.y), Circle(O,r2))[0]; pair[] points={X,O,Y,Z};
filldraw(Circle(O,r2),mediumgray,black); filldraw(Circle(O,r1),white,black);
dot(points); draw(X--Y--O--cycle--Z);
label("

O

",O,SSW,fontsize(10pt)); label("

Z

",Z,SW,fontsize(10pt)); label("

Y

",Y,N,fontsize(10pt)); label("

X

",X,NE,fontsize(10pt));
defaultpen(fontsize(8pt));
label("

c

",midpoint(O--Z),W); label("

d

",midpoint(Z--Y),W); label("

e

",midpoint(X--Y),NE); label("

a

",midpoint(X--Z),N); label("

b

",midpoint(O--X),SE);[/asy]

Problem Statement