This is a variation of Pythagorean quilts.
I will make it short, this time.
Pythagoras's theorem also works for triangles. This leads to the following variation:
Dissect the triangles of size 5 and 12 and use the pieces to recompose a triangle of size 13.
You are allowed to cut only along triangle boundaries, i.e. the gray lines. Straight cuts are not required, you can cut any twisted path you like. The resulting pieces must be connex by sharing full sides, not only by touching corners.
Unlike for the pythagoran quilts, you are allowed to rotate or flip the pieces. In fact, the count of "up" and "down" triangles doesn't match on both sides so a flip or a rotation will be necessary.
As usual, you have to minimize the number of pieces.