Rearranging the pieces creates different hypotenuses so they have different slopes....thus...changing the amount of space inside them.
The slope of the hypotenuse in the first example is not consistent with that of the second. The slight upward tick toward the far right side encompasses the area of the missing block once the figures are arranged to have a constant slope.
The Gentlemen were correct. The two triangles have different slopes. If you take a straight edge to the top one and then the bottom one you will see bulges in opposite directions. The slight differences account for the hole.