2018 AMC 12B Problems/Problem 8
Problem
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
For each 
note that the length of one median is 
Let 
be the centroid of 
It follows that 
Two shapes of namely 
and 
with their respective centroids 
and 
are shown below:
![[asy] /* Made by MRENTHUSIASM */ size(200); pair O, A, B, C1, C2, G1, G2, M1, M2; O = (0,0); A = (-12,0); B = (12,0); C1 = (36/5,48/5); C2 = (-96/17,-180/17); G1 = O + 1/3 * C1; G2 = O + 1/3 * C2; M1 = (4,0); M2 = (-4,0); draw(Circle(O,12)); draw(Circle(O,4),red); dot("$O$", O, (3/5,-4/5), linewidth(4.5)); dot("$A$", A, W, linewidth(4.5)); dot("$B$", B, E, linewidth(4.5)); dot("$C_1$", C1, dir(C1), linewidth(4.5)); dot("$C_2$", C2, dir(C2), linewidth(4.5)); dot("$G_1$", G1, 1.5*E, linewidth(4.5)); dot("$G_2$", G2, 1.5*W, linewidth(4.5)); draw(A--B^^A--C1--B^^A--C2--B); draw(O--C1^^O--C2); dot(M1,red+linewidth(1),UnFill); dot(M2,red+linewidth(1),UnFill); [/asy]](http://latex.artofproblemsolving.com/0/b/e/0be48fd18dce9f5eb975e87901bc6d9889fb0baf.png)
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 
~MRENTHUSIASM
See Also
| 2018 AMC 12B (Problems • Answer Key • Resources) | |
| Preceded by Problem 7 |
Followed by Problem 9 |
| 1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 | |
| All AMC 12 Problems and Solutions | |
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