smc

Question

Line segment 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 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?

Accepted Answer

For each ABC, note that the length of one median is OC=12. Let G be the centroid of ABC. It follows that OG=13 OC=4. As shown below, ABC_1 and ABC_2 are two shapes of ABC with centroids G_1 and G_2, respectively: FIGURE Therefore, point G traces out a circle (missing two points) with the center O and the radius OG, as indicated in red. To the nearest positive integer, the area of the region bounded by the red curve is OG^2=1650.