Geometry

If you look carefully, you can see that the pieces do not line up on
the larger triangle. The big triangle has a height/length ratio
of 5/13 (0.3846153).

On the top picture: the left sub-triangle is 3/8 (0.375) and the right
sub-triangle is 2/5 (0.4). The green diamond-shaped wedge is the
area deficit from the true hypontenuse. When the pieces are
switched on the bottom picture, the green wedge is now the area excess
from the true hypontenuse.

Area of big triangle = 0.5*5*13 = 32.5

The area of individual top pieces = 15 + (0.5*2*5) + (0.5*3*8) = 32

The area of the individual bottom pieces + green square = 33

The area of the wedge on top + area of wedge on bottom = 1. The
wedge for each triangle is 0.5