AMC 12 Mock Exam 5 Problem 7

Tangent segments from a vertex to the circle are equal. Going around the trapezoid, each side is the sum of two tangent segments, and every tangent segment is used once by a base and once by a leg. Hence the two legs together equal the two bases together: $2\ell=8+18$, so each leg is $\ell=13$.

Dropping perpendiculars from the ends of the shorter base cuts off two right triangles, each with horizontal leg $\frac{18-8}{2}=5$ and hypotenuse $13$. So the height is $\sqrt{13^2-5^2}=12$. The circle is tangent to both parallel sides, so its diameter equals the height, and the radius is $\boxed{6}$.