Example 3. As shown in Figure 2, given the line $\mathrm{y}=2 \mathrm{x}$ and point $\mathrm{M}(6, 2)$, try to find points $A$ and $B$ on the $x$-axis and the line $y=2 x$ respectively, such that the perimeter of $\triangle A \bar{B} M$ is minimized, and find the minimum perimeter.
Solving directly is difficult. If we take the symmetric points of point M with respect to the $x$-axis and the line $\mathrm{y}=2 \mathrm{x}$, denoted as $\mathrm{M}^{\prime}$ and $\mathrm{M}^{\prime \prime}$, respectively, and find their coordinates, the problem then becomes finding the distance between them and the intersection points of their line with the $x$-axis and $y=2 x$. (Solving for
$$
\begin{array}{l}
\mathrm{A}(4,0), \mathrm{B}\left(\frac{4}{3}, \frac{8}{3}\right), \triangle \mathrm{ABM} \text { perimeter }=\mathrm{M}^{\prime} \mathrm{M}^{\prime \prime} \\
=8 \sqrt{2} .)
\end{array}
$$