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 $C?$

$9\pi$

$12\pi$

$18\pi$

$24\pi$

$27\pi$

Answer: B

Concepts:equilateral triangle circumcircle, circumcenter, and circumradius law of sines

Difficulty rating: 1610

Solution:

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.

2021 AMC 12B Fall Problem 9

Answer: B

Difficulty rating: 1610

Solution:

Problem 9 in Other Years