 Puzzler

When it all becomes just too much to bear, scroll to the bottom of the page for the answer It is difficult to see in the picture as drawn with its thick boundary lines but if you redraw the puzzle in Illustrator with narrower lines it becomes somewhat easier to see. The answer is that neither the top or bottom composite figures are true triangles. The pseudo hypotenuse in the upper figure bulges downwards a very small amount while the pseudo hypotenuse in the lower figure bulges upwards by a similarly small amount. The difference in the areas covered by these two quadrilaterals is exactly enough to amount to one grid square. Compare corresponding points along the pseudo hypotenuse of each figure to nearby grid intersections to see this effect. You can prove this analytically by using a bit of trig to compute three angles for each of the figures using the grid square lengths of the sides as known data:

```
1) The hypotenuse angle of the dark green triangle
= atan (2/5) = atan (0.4) = 21.801 deg
2) The hypotenuse angle of the red triangle
= atan (3/8) = atan (0.375) = 20.556 deg
3) The hypotenuse angle of the total pseudo triangle (not drawn)
= atan (5/13) = atan (0.385) = 21.038 deg

For a true triangle, all three angles must necessarily be identical.
For the figures shown the three angles are all different.

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