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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:
We reflect $$\overline{B C}$$ about the $$y$$-axis to get $$\overline{B C^{\prime}}$$.
 We reflect $$\overline{C D}$$ 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
$$A B+B C+C D =A B+B C+C D^{\prime} $$
$$=A B+B C^{\prime}+C^{\prime} D^{\prime \prime} $$
$$=A D^{\prime \prime} $$
$$=\sqrt{((3-(-7))^{2}+(5-(-5))^{2}.} $$
$$= 10 \sqrt{2} $$

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:
We reflect $\overline{B C}$ about the $y$ -axis to get $\overline{B C^{\prime}}$ .
We reflect $\overline{C D}$ 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
$A B+B C+C D =A B+B C+C D^{\prime}$
$=A B+B C^{\prime}+C^{\prime} D^{\prime \prime}$
$=A D^{\prime \prime}$
$=\sqrt{((3-(-7))^{2}+(5-(-5))^{2}.}$
$= 10 \sqrt{2}$

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