Question

A sphere with center $O$ has radius $6.$ A triangle with sides of length $15, 15,$ and $24$ is situated in space so that each of its sides is tangent to the sphere. What is the distance between $O$ and the plane determined by the triangle?

$2\sqrt{3}$

$4$

$3\sqrt{2}$

$2\sqrt{5}$

$5$

Solution:

The sphere intersects the triangle's plane in a circle of radius $\sqrt{36 - d^2},$ where $d$ is the distance from $O$ to the plane. Since each side is tangent to the sphere, this circle is the triangle's incircle.

The triangle has area $\tfrac{1}{2} \cdot 24 \cdot 9 = 108$ and semiperimeter $27,$ so its inradius is $\dfrac{108}{27} = 4.$

Thus $\sqrt{36 - d^2} = 4,$ giving $d^2 = 20$ and $d = 2\sqrt{5}.$

Thus, the correct answer is D.

2019 AMC 12A Problem 18

Solution:

Problem 18 in Other Years