I am damn right confused!! What is wrong with this figure … geometrically, an area of 64 sq. units is now converted into 65 sq. units. How!?!

Got any bright ideas on this? I do.

I think it is violating the Basic Proportionality Theorem (BPT). Since it is assuming that 3/8 = 5/13 (the colored triangle being similar to the bigger half of the rectangle). Which is obviously incorrect, so the rectangle cannot be formed like so. Whew! Thank god for the constraints of geometry!!

A precise (and a slightly lengthy) answer is here.