2018 AMC 10 B
Complete problem set with solutions and individual problem pages
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? (2018 AMC 10B Problem, Question#12)
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Let , . Therefore, lies on the circle with equation . Let it have coordinates . Since we know the centroid of a triangle with vertices with coordinates of , , is , the centroid of is . Because , we know that , so the curve is a circle centered at the origin. Therefore, its area is .
We know the centroid of a triangle splits the medians into segments of ratio , and the median of the triangle that goes to the center of the circle is the radius (length ), so the length from the centroid of the triangle to the center of the circle is always . The area of a circle with radius is , or around .
