Answer to amazing triangle problem - the time has finally come to reveal the answer. (Over 100 emails later)

This was one of the best brainteasers I have ever seen that will crumble to the analytical process. And I am happy to report that fully 50% of the emails I received had the correct answer. Rather than bore you with my own explanation, I will quote some of the more enlightening emails I received.

A new type of geometry?

A quick glance showed you've created two new animals (1) concave triangles (actually 4 sided figures) (2) convex triangles and everyone knows that a convex triangle has more area than a concave one (slopes .375 and .4 aren't linear when combined)

good one, later,

Barry

The heart of all brainteasers!

Tell me, would trust your moistened finger tips to judge the difference between 3.3 and 5 volts (or 100 and 110)? Why do you trust your naked eyeballs to judge small differences in angles?

Assuming that all points lie on the grid lines shown (an important assumption that you did not state), the green triangle has proportions (height:width) of 2:5, the red triangle 3:8, and the overall figure (which isn't a triangle) 5:13. None of these triangles are similar to each other (remember similar trangles from plane geometry?). Note the blue triangle in the attached figures.

This is a variation on some common brainteasers, all based on getting people to assume things that are not so. In this case the angles of the triangles are close (but not the same), many people automatically assume they are indeed the same.

Edward A. Gardner

What you see is not always what you get!

Hmm, not so much a geometry problem as an optical illusion. Problem is that neither of the two figures is actually a triangle. The hypotenuse on each is bent. That can be easily seen by adding the areas of the individual pieces, which comes out to 32 square units. If the combined figure was actually a (right) triangle the area would be 1/2 the base x height, or 1/2 x 5 x 13, which is 32.5. So there is a half square discrepancy on the area above and below the diagonal. Together they show up above the bent 'hypotenuse' on the top figure and below on the bottom.

Sven

And my favorite, not because he called me by name, but this is exactly the process I went through to find the solution.

Darren,

Very sneaky. I calculated the composite area of the triangle and got 32.5. Then I added the sum of the individual shapes and got 32.

Huh?

So then I calculated the composite area of the lower triangle and got 31.5.

Double "huh"!

My suspicions were drawn to the triangles, they must not all combine into a REAL triangle. That had to be where the discrepancy was. Looking closely at the upper shape, I noticed that the shape wasn't quite right. Checking the proportionality of the two triangles I discovered my first SOLID clue:

small triangle: 5 across, 2 high
large triangle: 8 across, 3 high

If the angles of the triangles were identical, they would be proportional. The large one would have to be 7.5 across, 3 high.

That's where the area went! The upper and lower "triangles" are not identical shapes.

Stumped me for a few minutes!

Kurt Gunther

The plan is to do more of these, so feel free to email me with your favorite 'teaser, if it's good enough, we will put it up for all to see.

Darren

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