2018 AMC 12B Problem #8

Geometry Level 2

Line segment A B \overline{AB} s a diameter of a circle with A B = 24. AB = 24. Point C , C, not equal to A A or B , B, lies on the circle. As point C C moves around the circle, the centroid (center of mass) of A B C \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?

This problem is part of the 2018 AMC 12B .
38 63 50 75 25

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Marta Reece
Feb 16, 2018

The point has to be on a line connecting C to the middle of AB, that is to the center O of the circle.

It also has to be located 1 3 \frac13 of the distance OC from O.

The curve is therefore a circle centered at O with a radius 1 3 \frac13 of the radius of the original circle, that is r = 1 2 1 3 24 = 4 r=\frac12\cdot\frac13\cdot24=4 .

The area of this circle is then A = 16 π 50 A=16\pi\approx\boxed{50} .

0 pending reports


Problem Loading...

Note Loading...

Set Loading...