# Reflection Box

Geometry Level 3

A beam of light is emitted within a rectangular box from point $$(x,y) = (3,\frac{1}{2})$$ at an angle of $$45^{\circ}$$ with respect to the horizontal. The coordinates of the four box corners are $$(0,0),(4,0),(4,2),$$ and $$(0,2)$$, as shown in the figure.

The beam of light reflects off of the sides of the box such that the horizontal velocity is reversed when the beam strikes a vertical side, and the vertical velocity is reversed when the beam strikes a horizontal side.

Determine the $$x$$ coordinate of the location at which the beam first strikes the bottom side $$(y=0)$$.

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