7. As shown in Figure 12, six congruent equilateral triangles form $口 A B C D$, and the length of line segment $A C$ is 10. Find the length of line segment $X Y$.

保留源文本的换行和格式，直接输出翻译结果如下：

7. As shown in Figure 12, six congruent equilateral triangles form $口 A B C D$, and the length of line segment $A C$ is 10. Find the length of line segment $X Y$.
7. $\frac{5}{6}$.

As shown in Figure 16.
Figure 16

By symmetry, we have $A O=O C$.
In $\triangle C T_{3} O$, we have
$$
\frac{C Y}{C O}=\frac{C T_{4}}{C T_{3}}=\frac{1}{2} \Rightarrow C Y=\frac{1}{2} C O.
$$

In $\triangle C T_{3} M$, we have
$$
\frac{C X}{C M}=\frac{C T_{4}}{C T_{3}}=\frac{1}{2} \Rightarrow C X=\frac{1}{2} C M.
$$

By symmetry, we know
$$
\begin{array}{l}
C M=2 C O-A M=2 C O-C X \\
\Rightarrow C X=C O-\frac{1}{2} C X \\
\Rightarrow C X=\frac{2}{3} C O. \\
\text { Also, } X Y=C X-C Y \\
=\frac{2}{3} \mathrm{CO}-\frac{1}{2} \mathrm{CO}=\frac{1}{6} \mathrm{CO}, \\
C O=\frac{1}{2} \mathrm{AC}=5,
\end{array}
$$

Therefore, $X Y=\frac{5}{6}$.