In Michelson and Morley experiment, light is travelling from light source to partially silvered glass plate. The plate splits an incoming beam of light, and the two resulting beams continue in mutually perpendicular directions to the mirrors. Now, while the light is on its way from the plate to the mirror, the apparatus moves a distance $v t_1$, so the light must traverse a longitudinal distance $L+v t_1$, at the speed $c$. But from the plate to another mirror the light travels a distance along the hypotenuse of a triangle.
It is clear for me that the mirror (in a transverse direction) also must traverse a longitudinal distance $v t_1$. But I don't understand why should the light travel a distance along the hypotenuse of a triangle following the mirror. At the splitting moment, the beam receives a perpendicular direction (along with the cathetus toward the mirror location point at the splitting moment). At what moment and why does the direction of the beam change letting the beam to reach the mirror? Why doesn't beam continue with the initial direction (at the splitting moment) and why doesn't beam miss the mirror?
Please, help me to understand :)