AMC12 2018 B

AMC12 2018 B · Q8

AMC12 2018 B · Q8. It mainly tests Triangles (properties), Geometry misc.

Line segment $\overline{AB}$ is a diameter of a circle with AB = 24. Point C, not equal to A or B, lies on the circle. As point C moves around the circle, the centroid (center of mass) of $\triangle ABC$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?

线段 $\overline{AB}$ 是圆的直径，AB = 24。点 C（不等于 A 或 B）位于圆上。当点 C 在圆周上移动时，$\triangle ABC$ 的质心（质心）描出的是一条缺少两个点的闭合曲线。该曲线包围的区域面积最接近哪个正整数？

(A) 25 25

(B) 38 38

(D) 63 63

(E) 75 75

Answer

Correct choice: (C)

正确答案：（C）

Solution

Answer (C): Let $O$ be the center of the circle. Triangle $ABC$ is a right triangle, and $O$ is the midpoint of the hypotenuse $AB$. Then $\overline{OC}$ is a radius, and it is also one of the medians of the triangle. The centroid is located one third of the way along the median from $O$ to $C$, so the centroid traces out a circle with center $O$ and radius $\frac{1}{3}\cdot 12=4$ (except for the two missing points corresponding to $C=A$ or $C=B$). The area of this smaller circle is then $\pi\cdot 4^2=16\pi\approx 16\cdot 3.14\approx 50$.

答案（C）：设 $O$ 为圆心。三角形 $ABC$ 是直角三角形，且 $O$ 是斜边 $AB$ 的中点。则 $\overline{OC}$ 是一条半径，同时也是该三角形的一条中线。重心位于从 $O$ 到 $C$ 的中线上距离 $O$ 处的三分之一处，因此重心会描出一个以 $O$ 为圆心、半径为 $\frac{1}{3}\cdot 12=4$ 的圆（但对应 $C=A$ 或 $C=B$ 的两点缺失）。这个较小圆的面积为 $\pi\cdot 4^2=16\pi\approx 16\cdot 3.14\approx 50$。

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