2018 AMC 10B Problems/Problem 12
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
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
.
-tdeng
See Also
2018 AMC 10B (Problems • Answer Key • Resources) | ||
Preceded by Problem 11 |
Followed by Problem 13 | |
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