2018 AMC 12B Problem 8

Question

Line segment

\overline{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

Let

O

be the center; the centroid lies one-third of the way from

O

toward

C

As

C

moves on the circle of radius

12,

the centroid traces a circle of radius

\tfrac13\cdot12

Let

O

be the center of the circle. The centroid of

\triangle ABC

is the average of

A,

B,

and

C;

since

O

is the midpoint of

\overline{AB},

the centroid lies one-third of the way from

O

to

C.

As

C

traces the circle of radius

12,

the centroid traces a circle of radius

\tfrac13\cdot12=4.

Its area is

16\pi\approx50.

Thus, the correct answer is C.

Problem 8 in other years