2019 AMC 12A Problem 18

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.