Difference between revisions of "2018 AMC 12B Problems/Problem 8"

Revision as of 17:26, 18 June 2018

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

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

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 $\pi\cdot4^2=16\pi \approx \boxed{50}$ .

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

The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.

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 {{AMC12 box|year=2018|ab=B|num-a=9|num-b=7}}
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+[[Category:Intermediate Geometry Problems]]

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Difference between revisions of "2018 AMC 12B Problems/Problem 8"

Revision as of 17:26, 18 June 2018

Problem

Solution

See Also