A laser is placed at the point $(3,5)$. The laser beam travels in a straight line. Larry wants the beam to hit and bounce off the $y$-axis, then hit and bounce off the $x$-axis, then hit the point $(7,5)$. What is the total distance the beam will travel along this path?
$\textbf{(A) }2\sqrt{10} \qquad \textbf{(B) }5\sqrt2 \qquad \textbf{(C) }10\sqrt2 \qquad \textbf{(D) }15\sqrt2 \qquad \textbf{(E) }10\sqrt5$
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 $\overline{BC}$ about the $y$-axis to get $\overline{BC'}.$
We reflect $\overline{CD}$ about the $x$-axis to get $\overline{C'D'}$ with $D'=(7,-5),$ then reflect $\overline{C'D'}$ about the $y$-axis to get $\overline{C'D''}$ with $D''=(-7,-5).$

We obtain the following diagram:
[asy] /* Made by MRENTHUSIASM */ size(225);   int xMin = -9; int xMax = 9; int yMin = -7; int yMax = 7;  draw((xMin,0)--(xMax,0),black+linewidth(1.5),EndArrow(5)); draw((0,yMin)--(0,yMax),black+linewidth(1.5),EndArrow(5)); label("$x$",(xMax,0),(2,0)); label("$y$",(0,yMax),(0,2));  pair A = (3,5); pair B = (0,2); pair C = (2,0); pair D = (7,5); pair E = (-2,0); pair F = (7,-5); pair G = (-7,-5);  draw(A--B--C--D,red); draw(B--E,heavygreen+dashed); draw(C--F,heavygreen+dashed); draw(E--G,heavygreen+dashed); dot("$A(3,5)$",A,(0,2),linewidth(3.5)); dot("$B$",B,(-2,0),linewidth(3.5)); dot("$C$",C,(0,-2),linewidth(3.5)); dot("$D(7,5)$",D,(0,2),linewidth(3.5)); dot("$C'$",E,(0,-2),linewidth(3.5)); dot("$D'(7,-5)$",F,(0,-2),linewidth(3.5)); dot("$D''(-7,-5)$",G,(0,-2),linewidth(3.5)); [/asy]
The total distance that the beam will travel is 
\begin{align*} AB+BC+CD&=AB+BC+CD' \\ &=AB+BC'+C'D'' \\ &=AD'' \\ &=\sqrt{((3-(-7))^2+(5-(-5))^2} \\ &=\boxed{\textbf{(C) }10\sqrt2}. \end{align*}
~MRENTHUSIASM (Solution)
~JHawk0224 (Proposal)