2018 AMC 10B Problem 12

Problem 12 of 25IntermediateGeometry

Line segment

AB

is a diameter of a circle with

AB = 24.

Point

C,

not equal to

A

B,

lies on the circle. As point

C

moves around the circle, the centroid (center of mass) of

\triangle 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?

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Solution

Put the center

O

at the origin, so

A = (-12, 0)

and

B = (12, 0),

while

C

runs over the circle of radius

12.

Then

A + B = 0,

so the centroid is

\tfrac13(A + B + C) = \tfrac13 C.

C

circles,

\tfrac13 C

traces a circle of radius

\tfrac{12}{3} = 4

(minus the two points where

C = A

B

). Its area is

\pi \cdot 4^2 = 16\pi \approx 50.

Therefore, the answer is C.

Tagged: centroid · homothety · circle area