In the top figure the long hypotenuse is actually concave (bowing inward) while in the bottom figure the hypotenuse is convex (bowing outward) making the contained area of the lower triangle greater than that of the top.

but ill keep talking anyways

if simple theory doesnt work on you you can do as one person did somewhere above and prove it with simple geometry. he figured it out by calculating all dimensions looking for the inconsistency, and found it.. a different angle on the two triangle.

you cannot argue that you built the pieces and it still works, because that is 100% correct, you will accomplish the exact same effect. the illusion here isnt how it is drawn, its just taking a small area, hiding it over a long distance or focusing it in a small space, the small angle is just real hard to see with the eye, and will be only hidden better by trying to tape little pieces of cut out paper together. now if you cut these out v ery accurately, and large enough to work with, and very accurately traces the perimiter, rearanged them as shown, you would see a gap totaling the same area develop, spread out along what is a bent line. basically a waste of time thought, all you did was copy what was already shown on your screen.

some of you are making this way way too complicated, others are too stupid to understand the problem. the trick to this isnt math, geometry, trig, its simple theory. the area an object cannot dissapear by rearranging pieces of it, but the resulting shape can change. so the shape of the two triangles MUST be different. and it is.

the missing area is hidden along the length of the hypotenuse by the different angle created.

So, dude… if you can’t figure out why the apparently straight line isn’t really straight, you can pick your favorite ruler and see this for yourself. Or maybe they’re not that straight anymore? It’s a fact, not a theory. With trigonometry, you can (on theory) see exactly which portions of the elements form the missing square.

The red&green triangles appear to be similiar (to have equal angles), but actually their angles differ. If you assume all the sizes of the shapes (which form the whole picture) are determined exactly by the number of squares they occupy, you can easily calculate the angles of the red&green triangles, using trigonometry.

This is a mathematical proof that the hypotenuse of the top (whole) triangle is not actually a straight line, which means that this object is actually not a triangle, which means that its area cannot be calculated (accurately) with the “right-triangle-formula”, but rather “the-sum-of-its-parts” approach should be used.

The conundrum here is that, the point where the red&green triangles meet appears to be exactly on top of the grid, which gives the erroneous impression that their sides are well-defined. In reallity, this picture is ambiguous. If the grid represents the sizes accurately, then the sides of the red&green triangles can’t be measured with whole numbers (integers), but irrational numbers should be used instead.

So… my conclusion is that:
1) The “whole” triangle is not REALLY triangle (its hypotenuse isn’t straight line)
2) If you get a grid of squares and draw a STRAIGHT line in such a way that it forms the hypotenuse of a right triangle with sides 5 and 13, you’ll notice that this line doesn’t go through any of the grid intersection-points (it will come close, but not exactly over where the grid lines intersect)
3)This is one EXCELLENT visual illusion.

If you take the top shape and kind of fold it over on the bottom shape, so that the green triangles and the red triangles make a 2×5 rectangle and a 3×8 rectangle, respectively, then you can see this more clearly. You get a 5×13 rectangle with one piece missing, because the area of 5×13 is 65 and the areas of all the colored shapes only adds up to 64.

There is no bending, just dissimilar triangles.

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