Possible Duplicate:
How come 32.5 = 31.5?
Why does rearranging the pieces of this triangle illusion give a different area?
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0@InterestedQuest, @Sunil This was the dupe: http://math.sta$c$kexchange.com/questions/287/how-come-32-5-31-5 – 2011-03-07
4 Answers
If you count carefully, you'll see that the base is meant to be $13$ units long, while the height is $5$ units long. That means that the triangle on the top on the top figure, which has a height of $2$, should have a base of length $b$, where $\frac{b}{2} = \frac{13}{5}$ or $b = \frac{26}{5}$, longer than the $5$ units depicted.
Likewise, the bottom red triangle, with a base of size $8$ should have a height of length $h$, with $\frac{h}{8} = \frac{5}{13}$ or $h = \frac{40}{13}$, which is a little longer than the $3$ depicted.
So in fact, the "missing square" comes from misdrawing the pictures (or from having the individual figures drawn correctly, but the composed figures not being real triangles; the two inner triangles are not similar, though they "should" be).
Eye trick! Look at the angles formed where the red and green triangle meet.
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0Indeed the thick lines are what make the "angles different" - and which is what makes it sloppy. – 2011-03-01
It isn't true. See the Wikipedia page about this puzzle.
Take your credit card, driver's license, or some other readily available straightedge and put it against the hypotenuse. You'll find the composite shape is not a triangle, but a cleverly disguised irregular quadrilateral. The better, more mathy answers made sense once I got my brain away from the idea that there were triangles involved.
Nothing but a cheap trick designed to confound.