1. As shown in Figure 10, in rectangle $A B C D$, $A B=20$, $B C=10$. If points $M$ and $N$ are taken on $A C$ and $A B$ respectively, such that the value of $B M+M N$ is minimized, find this minimum value.
(1998, Beijing Junior High School Mathematics Competition)
As shown in Figure 10, construct the symmetric point $B^{\prime}$ of point $B$ with respect to line $A C$, and let $B B^{\prime}$ intersect $A C$ at point $E$. Draw $B^{\prime} N \perp A B$ at point $N$, and let $B^{\prime} N$ intersect $A C$ at point $M$. Then, $M$ and $N$ are the required points. The minimum value sought is $B^{\prime} N$.

From $S_{\triangle A B C}=\frac{1}{2} A B \cdot B C=\frac{1}{2} A C \cdot B E$, we get $B E=4 \sqrt{5}$.
Therefore, $B B^{\prime}=2 B E=8 \sqrt{5}$.
From $\triangle B^{\prime} N B \sim \triangle A B C$
$\Rightarrow \frac{B^{\prime} N}{A B}=\frac{B^{\prime} B}{A C} \Rightarrow B^{\prime} N=16$.