There are two solutions. One is very simple. The starting points of the pre-collision vector and the post-collision vector are put together directly by translation, which just forms an equilateral triangle. The third side is the momentum increment, with the magnitude of mv and the direction to the left.

In a slightly more complicated decomposition, the increment in the Y direction is zero, and the momentum components in the X direction are -0.5mv and 0.5mv, and the difference between them is -mv.