Answer:
$\frac{40}{21}$
Solution: Let the side length of $U V W X Y Z$ be $s$. We have $W Z=2 s$ and $W Z \| A B$ by properties of regular hexagons. Thus, triangles $W C Z$ and $A C B$ are similar. $A W V$ is an equilateral triangle, so we have $A W=s$. Thus, using similar triangles, we have
$$
\begin{array}{l}
\quad \frac{W C}{W Z}=\frac{A C}{A B} \Longrightarrow \frac{8-s}{2 s}=\frac{8}{5} \text {, } \\
\text { so } 5(8-s)=8(2 s) \Longrightarrow s=\frac{40}{21} \text {. }
\end{array}
$$
2. Let $A B C$ be a triangle with $A B=5, A C=8$, and $\angle B A C=60^{\circ}$. Let $U V W X Y Z$ be a regular hexagon that is inscribed inside $A B C$ such that $U$ and $V$ lie on side $B A, W$ and $X$ lie on side $A C$, and $Z$ lies on side $C B$. What is the side length of hexagon $U V W X Y Z$ ?