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$
Solution
Let $A=(3,5)$ and $D=(7,5)$. Reflecting the part of the path after a bounce in the mirror straightens the path: let $D'=(7,-5)$ be the reflection of $D$ in the $x$-axis, and $D''=(-7,-5)$ the reflection of $D'$ in the $y$-axis.
The broken path from $A$ to the $y$-axis to the $x$-axis to $D$ has the same length as the straight segment $AD''$.
A laser is placed at the point $(2,4)$. The beam travels in a straight line, hits and bounces off the $y$-axis, then hits and bounces off the $x$-axis, then hits the point $(4,4)$. What is the total distance the beam will travel along this path?
