52. On the side $AB$ of the rectangular trapezoid $ABCD (AB \perp BC)$, a semicircle (with the side as the diameter) is constructed, which touches the side $CD$ at point $K$. The diagonals of the trapezoid intersect at point $O$. Find the length of the segment $OK$, if the lengths of the bases of the trapezoid $ABCD$ are 2 and 3.
Answer: $O K=6 / 5$. Solution. Note that $C K=B C=2$ and $D A=D K=3$ by the properties of tangents drawn to a circle (see Fig. 9). Since triangles $B C O$ and $D A O$ are similar by two angles ($\angle C B O=\angle A D O$ as alternate interior angles), then $B O / O D=B C / A D=2 / 3$. By the proven fact, $C K / K D=2 / 3$. Therefore, by Thales' theorem, $B C \| O K$. Thus, triangles $D O K$ and $D B C$ are similar, and by the property of similar triangles, we have $O K / B C=D K / D C=3 / 5$, from which we get $O K=6 / 5$.