Explanation of a Geometry Paradox

Last month I wrote about an interesting geometry trick, that resulted in a rather strange picture. In the post I explained why it happened, but it wasn't completely clear to some of the readers. So I decided to explain it again in a more complete way.

The problem is the following picture:

In the previous post I just wrote that the slope was different. For me such explanation seemed complete so I just continued ot talk about importance of proofs. In this post I will try to explain what it means that the slope is different.
First of all, it should be obvious that the area of both triangles should be equal, because they are built from the some blocks. This means that the missing segment area is still in the picture, but in a different place. Lets look on the following illustration:



In this picture the D dot is moved slightly to create a change in the slope. It is easy to see that the second triangle (on the right) is identical to the first (on the left) but it has another triangle added to it. This is exactly was was done in the original picture. The tiles were rearranged in such a way that the line connecting the points A and C is no longer straight. If the distance that the dot D has moved is small enough we don't see this, and the triangles seem identical.
Lets calculate how much the dot needs to move in order to account for the missing segment area.
In our picture the size of the triangle is: AB=5, BC=13. Thus according to Pythagoras:

The missing area is a 1x1 square. If the area of the ADC triangle is also equal to 1, we get the following equation:

And therefore h (the height of the ADC triangle) is: h=1/7 of a square. I rounded the figures a bit but the result is close enough. The result seems large, but it is not. It is of course possible to see, but it is not immediately visible.

The only thing that remains, is to show that indeed in the original picture the slope is different, and that the rearrangement of the pieces creates an extra triangle on top of the original one. This is very simple to do. All you need is to download the image to your computer and open it in an image editing program that supports layers and transparency. Now, cut the bottom triangle and put it in a different layer. Place it above the top triangle and play with transparency. Or you can watch the video below:

The program used to edit the picture is Gimp. It is a free program, capable of replacing Photoshop. The only problem with it is that it is not very intuitive.

I hope this post answered the question. If not, let me know in the comments.