2018 AMC 12B Problem #8
Line segment s 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?
This problem is part of the 2018 AMC 12B .
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1 solution
The point has to be on a line connecting C to the middle of AB, that is to the center O of the circle.
It also has to be located 3 1 of the distance OC from O.
The curve is therefore a circle centered at O with a radius 3 1 of the radius of the original circle, that is r = 2 1 ⋅ 3 1 ⋅ 2 4 = 4 .
The area of this circle is then A = 1 6 π ≈ 5 0 .