The Disappearing Square Puzzle

You may have seen something like this before: take a look at the image below. You'll see that I've divided a triangle up into four smaller shapes.

Now I'm going to rearrange those pieces to form a new triangle. Ready?

Here's the question: Why is there a gap in the second triangle? What happened to that missing square?

Before you panic, thinking that I've magically destroyed the rules of mathematics and geometry, take a close look at the pictures. Can you figure out what happened?

Help Me Figure It Out!

Want a hint?

Are you sure?

Here's your Hint: Can you calculate the slope of the line that goes through the diagonal of the whole piece of graph paper? It's change in Y over change in X, or -13/5. Okay, what about the slope of the diagonal line that is part of the red triangle? Hmmm...that would be -8/3.

WAIT A MINUTE! Those aren't the same number! -13/5 does not equal -8/3! You know what that means, don't you? The dimensions of the red triangle aren't REALLY 8x3! And guess what! The dimensions of the purple triangle are not really 5x2!

I'll leave it to you to figure out the rest...