First note that if the line $l$ has slope $a$, then the reflection of $l$ with respect to any line which is parallel to one of the axes has the slope $-a$.

Now assume that at the start the ray has slope $a$. Then the ray always has the slope $\pm a$. Now at the starting point a line with slope $-a$ lies outside the rectangle so by symmetry at the opposite corner the line with slope $-a$ lies outside the rectangle. Hence the ray reaches the opposite corner with slope $a$.

Now assume that a light ray enters the rectangle from the opposite corner. By symmetry these two rays meet at the time $\frac{t}{2}$ in the center of the rectangle.