1. As shown in Figure 6, some segment lengths of the three rectangles are marked. Try to find the length of segment $X Y$.

 untranslated part:
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1. As shown in Figure 12, extending the sides of the rectangle forms a right triangle $\triangle ABC$. Given $AB=2+5+3=10, BC=4+6=10$
$$
\Rightarrow \angle BCA=\angle BAC=45^{\circ} \text{.}
$$

By the Pythagorean theorem,
$$
\begin{array}{l}
AC=10 \sqrt{2}, AD=2 \sqrt{2}, DX=\sqrt{2}, \\
CE=2 \sqrt{2}, EY=\frac{\sqrt{2}}{2} .
\end{array}
$$

Thus, $XY=AC-AD-DX-CE-EY$
$$
=10 \sqrt{2}-2 \sqrt{2}-\sqrt{2}-2 \sqrt{2}-\frac{\sqrt{2}}{2}=\frac{9 \sqrt{2}}{2} \text{.}
$$