[ Projections of the bases, sides, or vertices of a trapezoid ] The Law of Sines

A quadrilateral $K L M N$ is inscribed in a circle of radius $R$, $Q$ is the point of intersection of its diagonals, $K L = M N$. The height dropped from point $L$ to side $K N$ is 6, $K N + L M = 24$, and the area of triangle $L M Q$ is 2. Find the sides of the quadrilateral and the radius of the circle $R$.
Prove that the given quadrilateral is an isosceles trapezoid. Denote its bases as \( x \) and \( y \). Use the similarity of triangles \( L Q M \) and \( N Q K \).

## Answer

\( L M = 4, K N = 20, K L = M N = 10 ; R = 5 \sqrt{5} \).