2018 AMC 12B Problems/Problem 8
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?
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
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 .
First we can draw a few conclusions from the given information. Firstly we can see clearly that the distance from the centroid to the center of the circle will remain the same no matter is on the circle. Also we can see that because the two legs of every triangles will always originate on the diameter, using inscribed angle rules, we know that . Now we know that all triangles will be a right triangle. We also know that the closed curve will simply be a circle with radius equal to the centroid of each triangle. We can now pick any arbitrary triangle, calculate its centroid, and the plug it in to the area formula. Using a -- triangle in conjunction with the properties of a centroid, we can quickly see that the length of the centroid is now we can plug it in to the area formula where we get .
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