8,9 |

Given a trapezoid $A B C D$ with bases $A D=3$ and $B C=18$. Point $M$ is located on diagonal $A C$, such that $A M$ : $M C=1: 2$. A line passing through point $M$ parallel to the bases of the trapezoid intersects diagonal $B D$ at point $N$. Find $M N$.
Let $K$ be the point of intersection of the given line with the side $A B$. By the theorem of proportional segments, $A K: K B = A M: M C = 1: 2$. Similarly, $B N: N D = B K: A K = 2: 1$. From the similarity of triangles $A K M$ and $A B C$, we find that $M K = B C \cdot A M / A C = 1 / 3 B C = 6$, and from the similarity of triangles $B K N$ and $B A D$, $N K = A D \cdot B N / N D = 2 / 3 A D = 2$. Therefore, $M N = M K - N K = 6 - 2 = 4$.

## Answer

4.