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# Triangle dissection problem

infinisa
Hello folks

Here is a fun paradox for you to solve:

How come I can rearrange the same pieces and get a unit square left over???

It's not TOO difficult to figure out.

Good luck & have fun.

P.S.: Solve this without doing an Internet search!
Afaceinthematrix
After thinking a while, it appears that the hypotenuses of the two triangles are not equal which means that the line of the overall hypotenuse of the entire figure is not a straight line - which creates an arc.

Edit:

After reading my explanation, I realized how poor it was. By equal, I meant the slopes of the two triangles are not equal. If my memory serves me right (I did this earlier today and I'm too lazy to go back and recheck), the slopes were 2/5 and 3/8.
infinisa
Hello Afaceinthematrix

Well spotted!

In fact this us a bit of a trick question, as the two "triangles" are not actually triangles at all, as their diagonals are not (quite) straight lines. Strictly speaking, they're quadrilaterals. The upper figure has its "diagonal" slightly bent in, whilst the lower figure has its "diagonal" slightly bent out. This hardly noticeable difference makes the figures differ in area by exactly 1 square unit.

If the figure were really a triangle, its slope would be 8/13.

As you correctly point out, the diagonal is actually made up of the diagonal of two triangles with slightly different slopes: 2/5 & 3/8.

These numbers don't happen by chance: they're numbers in the famous Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, ..., where each number is the sum of the previous two, and the ratio of successive numbers gets closer and closer to a limit - the golden ratio - (1+√5)/2, much used in painting.
Afaceinthematrix
Yes. It was pretty tricky realizing that it was a quadrilateral instead of a triangle. I think the only reason I spotted that was because I decided to count boxes!