Let $ABCD$ be a square with side length $1.$ A laser is located at vertex $A,$ which fires a laser beam at point $X$ on side $\overline{BC},$ such that $BX = \frac{1}{2}.$ The beam reflects off the sides of the square, until it ends up at another vertex; at this point, the beam will stop. Find the length of the total path of the laser beam.

LiIIiam0216 Jun 8, 2024

#1**0 **

Once the laser hits line BC, it will immediately be reflected to the upper right corner. This will make two right triangles, each with legs of 1 and 0.5. We can calculte the hypotenuse of each of these triangles.

\(1^2 + \frac{1}{2}^2 = h^2\)

\(4/4+1/4 = h^2\)

\(5/4 = h^2\)

\(h = \sqrt5/2\)

Since the laser's path is two of these identical hypotenuses, we double this about to get the length of **sqrt(5).**

Maxematics Jun 9, 2024