Question

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?

$25$

$38$

$50$

$63$

$75$

Solution:

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 Problem 12

Solution:

Problem 12 in Other Years