Fiendishly tricky geometry puzzle

Start with a piece of paper 8" x 8", as shown on the left.  Make the two cuts shown.

Then shift the big lower-left piece up and to the left and move the triangle from the upper left corner to the lower right one.

The area that used to be 8x8 = 64 square inches now appears to be 9x7 = 63 square inches.  Where is the fallacy?

I pondered this for way too long, and finally had to cut some physical pieces of paper to solve the paradox.

The puzzle was originally created by or published by the famous Sam Loyd; I found it in the Futility Closet.