2018 AMC 10B Problem 12

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

\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?

25

38

50

63

75

The centroid of

\triangle ABC

is

\tfrac13(A + B + C),

and

A + B

is twice the center

As

C

moves on a circle of radius

12,

the centroid moves on a circle of one third the radius

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.

As

C

circles,

\tfrac13 C

traces a circle of radius

\tfrac{12}{3} = 4

(minus the two points where

C = A

or

B

). Its area is

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

Therefore, the answer is C.

Problem 12 in other years