I'll Tell You How This Can Be True.

[mbarrick]

I've seen this "paradox" on the web many times now in more than one language and I've yet to see anyone posting the solution. The fact that it keeps getting reposted leads me to believe that people actually think this is a genuine paradox when in fact it doesn't take more than grade 8 math to figure out what is "wrong" (although, admittedly, it isn't obvious at first glance).

The apparent larger triangle made up of the component shapes in the top and the bottom is five squares high and thirteen squares long.

** start grade 8 math **

The area of a right-angle triangle is one half of the area of the rectangle with the same length sides.

So:

(13 * 5 ) / 2 = 32.5

Now look at the individual shapes:

The green triangle: (5 * 2 ) / 2 = 5
The red triangle: (8 * 3) / 2 = 12
The yellow "L" shape has 7 squares.
The green "L" shape has 8 squares.

5 + 12 + 7 + 8 = 32

The area of the top shape is 32 squares, not the 32.5 is should be if it was a right angle triangle.
Likewise the area of the bottom shape, with the "paradoxical" extra square is 33 squares, not the 32.5 is should be if it were a right angle triangle.

Neither shape is a right angle triangle.

The small green triangle and the large red triangle in the two shapes do not have the same proportions.

5:2 ≠ 8:3

Using a common denominator to show this:

(5 * 3) : (2 * 3) ≠ (8 * 2) : (3 * 2)
15:6 ≠ 16:6

Obviously the angles in a 15:6 triangle are not going to be the same as a 16:6 triangle.

** end grade 8 math **

Now if you want to get into the heady world of *gasp* high-school algebra to get into more detail:

The 13 x 5 triangle, using Pythagoras (a² + b² = c²), has a hypotenuse of 13.928 units and (simple trigonometry) interior angles of 21.04°, 90°, and 68.96°.
The small green 5 x 2 triangle has a hypotenuse of 5.385 units and interior angles of 21.8°, 90°, and 68.2°.
The small red 8 x 3 triangle has a hypotenuse of 8.544 units and interior angles of 20.56°, 90°, and 69.44°.

We have a very obtuse triangle with sides that are 5.385, 8.544 and 13.928 units long and interior angles of 178.76°, 0.48°, and 0.76° "left over".

You'll find this "leftover" happens to have an area of one half square.

The upper shape "dents in" below the true hypotenuse of the 13 x 5 triangle by the dimensions of this obtuse triangle, making it "short" half a square in area.

Conversely the lower shape "bulges" above the true hypotenuse of the 13 x 5 triangle by the same amount, making it "over" half a square in area.

Comments

The visual obviousness

[(Anonymous)]

This is really, really late in comparison to the original entry date, but as I was checking out your site out I saw this, which I had actually never seen before, and I was puzzled. Though after reading your detailed explanation, I felt like it wasn't enough - surely you can prove they got the area of so many squares covered, but how would you hide it in a seemingly exact replica (apart from the bulging and denting), and that's when I noticed that it is quite obviously represented in the image itself, if you pay attention.

The seventh vertical line crosses the hypotenuse in a corner on the second triangle, so that the squares of the yellow shape does not extrude. On the first triangle, the hypotenuse does not even touch that same corner. Likewise, the tenth vertical line crosses the hypotenuse earlier (on it's way down) on the second triangle that the first - also here to make space for the yellow shape.

Naturally, the mathematical explanation is of course the best, but for anyone that skipped maths completely in school, a more graphical description might be more explanatory.

.. oh, and just a by the way, instead of making another comment, jolly good work with that Pedobear mascot. That surely got you some attention, eh? *winks*

Re: The visual obviousness

[mbarrick.livejournal.com]

You're quite right. All it really takes is looking closely, and just such an observation was antecedent to doing the math for me. But since you're here as a result of the Pedobear incident, you'll agree that careful observation really isn't some people's strong point ;-)