7. Line segments $A B, C D$ are between two parallel planes $\alpha$ and $\beta$, $A C \subset \alpha, B D \subset \beta, A B \perp \alpha, A C=B D=5$, $A B=12, C D=13, E, F$ divide $A B, C D$ in the ratio $1: 2$. Find the length of line segment $E F$.
7. Draw a perpendicular line from $D$ to plane $\alpha$, intersecting $\alpha$ at $H$. From $\alpha / / \beta, A B \perp \alpha$, we get $D H \cong A B$. Connect $A H, H C$. In the right triangle $\triangle D H C$, $D H=12, D C=13, D H \perp H C$, thus $H C=5, A H=5, \angle H C A=60^{\circ}$.

Clearly, the projection of $E$ on $\alpha$ is $A$, and the projection of $F$ on $\alpha$ is the trisection point $P$ of $C H$, i.e., $C P=\frac{1}{3} C H=\frac{5}{3}$. Using the cosine rule, we find $A P=\frac{5}{3} \sqrt{7}$. Therefore, $E F=\frac{5}{3} \sqrt{7}$.