2021 AMC 12B Fall Problem 9

Question

Triangle

ABC

is equilateral with side length

6.

Suppose that

O

is the center of the inscribed circle of this triangle. What is the area of the circle passing through

A,

O,

and

C?

9\pi

12\pi

18\pi

24\pi

27\pi

In an equilateral triangle the incenter coincides with the circumcenter

In triangle

AOC,

angle

AOC = 120^\circ

and

AC = 6

; apply the law of sines

For an equilateral triangle,

O

is also the circumcenter, so

OA = OC = \dfrac{6}{\sqrt3} = 2\sqrt3.

The central angle

\angle AOC = 120^\circ.

In triangle

AOC,

side

AC = 6

is opposite the

120^\circ

angle, so the circumradius

R'

of this triangle satisfies

2R' = \dfrac{6}{\sin 120^\circ} = 4\sqrt3,

giving

R' = 2\sqrt3.

The area of the circle is

\pi (2\sqrt3)^2 = 12\pi.

Thus, the correct answer is B.

Problem 9 in other years