In triangle top four figures that have to be repositioned to form the "triangle" without a unit square.
How to explain this?
The three points along what looks like a hypotenuse of a big right triangle are actually not collinear. So since the "big triangle" is not really one right triangle the usual area formula doesn't apply.
Note: in the top "triangle" the three points are $(0,0),(8,3),(13,5)$, so that IF they were on a line the slopes 5/13 and 3/8 would have to match. But they don't.
ADDED: In the top figure, a careful diagram shows the "hypotenuse" is made of two segments which actually bend outwards a little, and the bottom figure's "hypotenuse" is also two segments that bend inwards a little. This accounts for the difference in apparant areas.