......................

If you notice, the whole thing is not a triangle. You can check the slopes of the two triangle hypoteni:

One of them is 3/8 = 0.375

The other is 2/5 = 0.4

The missing area in the second one is compensated for on top, but you can't see it because the slopes are so close.

-Uncool-

......................

 

There's still something more to this. If you find the area of each region individually, it adds up to 32. But when you find the area of the triangle as a whole, it comes to 32.5. Is it possible to split up a triangle the way they did and have integers as the lengths of the regions?

Its' not that hard, the hypothenuse of the triangle is not a straight line, just hold a ruler against it...

There's still something more to this. If you find the area of each region individually, it adds up to 32. But when you find the area of the triangle as a whole, it comes to 32.5. Is it possible to split up a triangle the way they did and have integers as the lengths of the regions?

There's nothing more. The total area of all shapes is 32 squares when counted individually. The total area is 32.5 for the whole "triangle" if you use the formula 1/2*(base)*(height). But as has already been pointed out, the total "triangle" is not a triangle. It has a bent "hypotenuse," so you cannot use the latter formula. The correct area is 32 squares.