How's that again?

[ChristinaS]

Check this out:

[GSV3MiaC]

Re: How's that again?

The top 'triangle' is actually concave along what purports to be the hypotenuse .. the bottom one is convex, hence the extra space inside (they are really not triangles - just close!)

The green triangle and the red one are not similar triangles - the green one is 2, 5, sqrt(29), whereas the red one is 3, 8 (should only be 7.5 to match the green one) and sqrt (73).

The overall 'triangle' pretends to be 5, 13, sqrt(194) but you'll note that sqrt(194) is just a tiny bit smaller than sqrt(29)+sqrt(73) .. very subtle difference, but enough to make the hypotenuse not straight!

[ChristinaS]

Re: How's that again?

LOL, yes, something like that. Here's the explanation in some other brainy guy's words (I dare say it's the same - not gonna challenge either ):

Old one... The hypothenuses of the red and dark green triangles do not form a straight line. To form a straight line, the triangles need to be similar - thus, have equal corresponding angles and equal ratios of corresponding sides. Comparing ratios of corresponding cathetes, 2/3 is not equal to 5/8. To compare angles, we can use the tangent of one of the angles which, in a right triangle, is calculated by the ratio of the angle's opposing cathete and the adjacent cathete. In the red triangle that ratio is 3/8, in the dark green triangle the ratio is 2/5. Both methods prove that the triangles are not similar. Therefore, the two hypothenuses don't form a line (180 degrees between them); the actual angle is 181.245 degrees, but our eyes mislead us and approximate the difference as non existant (nice optical illusion, isn't it). These two hypothenuses form veeery slim and long triangles in both arrangements; the areas of these two triangles add up to exactly one square, which proves that the insignificant deviation of 1 degree and almost 15 minutes from a straight line can have significant results in computing the (wasted) area.

Pity, when I was doing my schooling we didn't study fun stuff like this (and me a math major too :o ) :D