AMC10 2004 B

AMC10 2004 B · Q12

AMC10 2004 B · Q12. It mainly tests Triangles (properties), Circle theorems.

An annulus is the region between two concentric circles. The concentric circles in the figure have radii $b$ and $c$, with $b > c$. Let $OX$ be a radius of the larger circle, let $XZ$ be tangent to the smaller circle at $Z$, and let $OY$ 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?

环形区域是两个同心圆之间的区域。图中的同心圆半径为$b$和$c$，其中$b > c$。设$OX$是大圆的一个半径，$XZ$在$Z$点与小圆相切，$OY$是大圆包含$Z$的半径。设$a = XZ$，$d = YZ$，$e = XY$。环形区域的面积是多少？

(A) $\pi a^2$ $\pi a^2$

(B) $\pi b^2$ $\pi b^2$

(D) $\pi d^2$ $\pi d^2$

(E) $\pi e^2$ $\pi e^2$

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) The area of the annulus is the difference between the areas of the two circles, which is $\pi b^2-\pi c^2$. Because the tangent $XZ$ is perpendicular to the radius $OZ$, $b^2-c^2=a^2$, so the area is $\pi a^2$.

（A）圆环的面积等于两个圆面积之差，即 $\pi b^2-\pi c^2$。由于切线 $XZ$ 垂直于半径 $OZ$，有 $b^2-c^2=a^2$，因此圆环的面积为 $\pi a^2$。

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