Question

线段 $AB$ 是一个圆的直径，且 $AB = 24$ 。点 $C$ 在圆上，但不等于 $A$ 或 $B$ 。当 $C$ 绕圆运动时， $\triangle ABC$ 的重心描出一条少了两个点的闭曲线。四舍五入到最接近的正整数，这条曲线围成区域的面积是多少？

Line segment $AB$ 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?

$25$

$38$

$50$

$63$

$75$

答案：C

知识点：重心位似圆面积

难度评级：1530

解答：

把圆心 $O$ 放在原点，则可取 $A = (-12, 0)$ 、 $B = (12, 0)$ ，而 $C$ 在半径为 $12$ 的圆上运动。因为 $A + B = 0$ ，重心为 $\tfrac13(A + B + C) = \tfrac13 C$ 。当 $C$ 绕圆运动时， $\tfrac13 C$ 描出半径为 $\tfrac{12}{3} = 4$ 的圆，少掉 $C = A$ 或 $B$ 时的两个点不影响面积。面积为 $\pi \cdot 4^2 = 16\pi \approx 50$ 。正确答案是 C。

Put the center $O$ at the origin, so $A = (-12, 0)$ and $B = (12, 0),$ while $C$ runs over the circle of radius $12.$ Then $A + B = 0,$ so the centroid is $\tfrac13(A + B + C) = \tfrac13 C.$ As $C$ circles, $\tfrac13 C$ traces a circle of radius $\tfrac{12}{3} = 4$ (minus the two points where $C = A$ or $B$ ). Its area is $\pi \cdot 4^2 = 16\pi \approx 50.$ Therefore, the answer is C.

2018 AMC 10B 第 12 题

答案：C

难度评级：1530

解答：

其他年份的第 12 题