smc

Let $A=(3,5)$ and $D=(7,5).$ Suppose that the beam hits and bounces off the $y$-axis at $B,$ then hits and bounces off the $x$-axis at $C.$ When the beam hits and bounces off a coordinate axis, the angle of incidence and the angle of reflection are congruent. Therefore, we straighten up the path of the beam by reflections: 1. We reflect $\overline{BC}$ about the $y$-axis to get $\overline{B{C}^{\prime }}.$ 2. We reflect $\overline{CD}$ about the $x$-axis to get $\overline{C^{\prime }{D}^{\prime }}$ with $D^{\prime }=(7,-5),$ then reflect $\overline{C^{\prime }{D}^{\prime }}$ about the $y$-axis to get $\overline{C^{\prime }{D}^{\prime \prime }}$ with $D^{\prime \prime }=(-7,-5).$ We obtain the following diagram: The total distance that the beam will travel is $$\begin{array}{rl}AB+BC+CD& =AB+BC+C{D}^{\prime }\\ & =AB+B{C}^{\prime }+C^{\prime }{D}^{\prime \prime }\\ & =A{D}^{\prime \prime }\\ & =\sqrt{((3-(-7){)}^2+(5-(-5){)}^2}\\ & =\boxed{10\sqrt{2}}.\end{array}$$