2018 AMC 12B Problems/Problem 8
Contents
Problem
Line segment is a diameter of a circle with . Point , not equal to or , lies on the circle. As point moves around the circle, the centroid (center of mass) of traces out a closed curve missing two points. To the nearest positive integer, what is the area of the region bounded by this curve?
Solution 1
For each note that the length of one median is Let be the centroid of It follows that
As shown below, and are two shapes of with centroids and respectively: Therefore, point traces out a circle (missing two points) with the center and the radius as indicated in red. To the nearest positive integer, the area of the region bounded by the red curve is
~MRENTHUSIASM
Solution 2
We assign coordinates. Let , , and lie on the circle . Then, the centroid of is . Thus, traces out a circle with a radius of the radius of the circle that point travels on. Thus, traces out a circle of radius , which has area , which is
See Also
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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