Triangles!

I’m going to mix it up on my blog a bit today. I found it odd how I am going to be graduating as a math major this upcoming year and that I do not have anything math related on my blog yet. I’ve decided to change that.

I’m sure most of you have heard of Pythagorean’s Theorem at some point during your schooling in high school.  The old a^2+b^2=c^2, where c is the hypotenuse of a right triangle, and a and b represent the two shorter sides of this right triangle. However, I bet most of you haven’t been shown a proof of this theorem and have just been told to except it unless you have taken math in some sort of post secondary education or you were curious enough one day to go and find a proof online or in a book. I’m going to introduce a neat proof today that I first saw in a video from one of my favorite YouTubers named Vihart. The reason why I found this proof particularly interesting is because it is not in the usual form of a mathematics proof which usually involves either writing lines and lines of explanation or manipulating axioms until the desired result is achieved. This proof is purely visual and very easy to understand.

Okay, so I’m no artist and all I had to work with was paint so bare with me…

In the above picture we see a square image with a smaller square in the interior of it bordered by 8 identical right triangles. If you wanted to make this with a piece of paper it is fairly easy. Just fold a square piece of paper in half twice lengthwise (not along the diagonal) and you should have a square that is 1/4 the size of the original square. Then fold this square in half along the diagonal to obtain a triangle. Finally, position this triangle so that the edges of the paper are at the bottom of your perspective of the paper and take the upper corner of the triangle and fold it downwards to create a crease which is parallel to the edges at the bottom of the triangle. It does not matter how far you fold the corner down it will still work. This step may seem confusing but try to visualize the words that I have written. After this you should obtain creases equivalent to the lines in the picture above.

We will label the hypotenuse of each of the 8 right triangles with an H, and the other two sides with B and L to denote big leg and little leg.

Now, we will remove the four triangles coloured in green. If you are playing along with the piece of folded paper there is no need to remove these pieces but simply fold them behind.

With these four green triangles removed we obtain a square (or we would if my paint skills were a bit better). One can see that each side of this square is the length of the hypotenuse of one of the right triangles. So, with some simple math we see that the area of this square is the length of the hypotenuse squared (or simply H^2).      

We will now refer back to this image but instead of removing the four green triangles we will remove four different triangles. We will remove the two upper triangles which share the hypotenuse labeled H in the picture as well as the two triangles on the left hand side of the square. With the folded paper you can make a small tear on the line labeled L and fold these four triangles behind.

With one quick glance at the above picture I am sure some of you clever ones will see that our proof is now complete. I will explain what happened during this last step. Since we removed four identical triangles during both of the previous steps this means that the area of the two different shapes created must be identical. One can see that the area of the above shape is the area of the small square in the top right of the image labeled with a 1 plus the area of the larger square which this small square is attached to labeled 2. The lengths of the sides of the small square are all L and the lengths of the sides of the larger square are all B. Thus, the area of the whole shape is B^2+L^2.
Finally, equating the areas of the two different shapes we obtain the Pythagorean Theorem, H^2=B^2+L^2.
QED


Cya guys next time.