2018 AMC 12B Problems/Problem 8

Problem

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 (insert triangle symbol) $ABC$ traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?

$\textbf{(A)} \indent 25 \qquad \textbf{(B)} \indent 32 \qquad \textbf{(C)} \indent 50 \qquad \textbf{(D)} \indent 63 \qquad \textbf{(E)} \indent 75$

Solution

Draw the Median connecting C to the center O of the circle. Note that the centroid is $\frac{1}{3}$ of the distance from O to C. Thus, as C traces a circle of radius 12, the Centroid will trace a circle of radius $\frac{12}{3}=4$ .

The area of this circle is $\pi\cdot4^2=16\pi \approx 50$ .

2018 AMC 12B (Problems • Answer Key • Resources)
Preceded by Problem 7	Followed by Problem 9
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All AMC 12 Problems and Solutions

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2018 AMC 12B Problems/Problem 8

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