AMC12 2018 B

AMC12 2018 B · Q4

AMC12 2018 B · Q4. It mainly tests Triangles (properties), Circle theorems.

A circle has a chord of length 10, and the distance from the center of the circle to the chord is 5. What is the area of the circle?

一个圆有一个长度为10的弦，从圆心到该弦的距离是5。求这个圆的面积。

(A) 25π 25π

(B) 50π 50π

(D) 100π 100π

(E) 125π 125π

Answer

Correct choice: (B)

正确答案：（B）

Solution

Answer (B): Let the chord have endpoints $A$ and $B$, and let $C$ be the center of the circle. The segment from $C$ to the midpoint $M$ of $\overline{AB}$ is perpendicular to $\overline{AB}$ and has length 5. This creates the $45-45-90$ triangle $CMB$, whose sides are 5, 5, and $CB=5\sqrt{2}$. Therefore the radius of the circle is $5\sqrt{2}$, and the area of the circle is $\pi\cdot(5\sqrt{2})^2=50\pi$.

答案（B）：设弦的端点为 $A$ 和 $B$，$C$ 为圆心。从 $C$ 到 $\overline{AB}$ 的中点 $M$ 的线段垂直于 $\overline{AB}$，且长度为 5。这样得到一个 $45-45-90$ 的三角形 $CMB$，其边长为 5、5，且 $CB=5\sqrt{2}$。因此圆的半径为 $5\sqrt{2}$，圆的面积为 $\pi\cdot(5\sqrt{2})^2=50\pi$。

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