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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.

 Jun 8, 2024
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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).

 Jun 9, 2024

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