Pythagorean theorem and its cause

Question

I'm in high school, and one of my problems with geometry is the Pythagorean theorem. I'm very curious, and everything I learn, I ask "but why?". I've reached a point where I understand what the Pythagorean theorem is, and I understand the equation, but I can't understand why it is that way. Like many things in math, I came to the conclusion that it is that way because it is; math is the laws of the universe, and it may reach a point where the "why" answers itself. So what I want to know is, is there an explication to why the addition of the squared lengths of the smaller sides is equal to the squared hypotenuse, or is it just a characteristic of the right triangle itself? And is math the answer to itself?

Thank you.

Answer 1

It is so because one can prove it. Several proofs can be found e.g. on Wikipedia. Of course one has to "believe" the axioms on which the proofs are based, so if you don't feel compelled to ackknowledge that e.g. "for every line in the plane and every point not on that line there exists one and only one line though this point that does not intersect the first line", the proofs won't work for you (but then also the statement itself may be wrong).

Answer 2

There are many proofs in math that tell us something is true. Some are pretty obvious such as the proof that "if $i$ and $j$ are integers and $i=2j$, then $i$ is even." That is almost the definition of "even" in fact.

Other proofs are very difficult and only a few people in the world understand them. Fermat's Last Theorem is a good example.

So "why" something is true depends on both how difficult the explanation is and how much training and experience and intelligence the reader has to understand that explanation. Fortunately, the Pythagorean Theorem is fundamental enough that most high schoolers can understand it, perhaps after some dedication. The picture given in a comment above does a very nice job of making it straightforward to see "why" it's true.

Note: This type of "picture proof" should always be viewed with a little suspicion since there are "false proofs" that use pictures that at-first seem convincing but have subtle errors. This one, however, is completely legit.

Answer 3

I love this question! It is deeply connected to coordinate systems and plane geometry. First, consider the relationship of the hypotenuse to a circle. Here I've drawn the 3-4-5 right triangle with the hypotenuse as the radius of the circle.

The length of the radius is 5, no matter how you look at it. And in fact, you can represent the coordinates in the traditional (x,y) coordinates as drawn, but any two perpendicular lines can form a "basis" for a coordinate system. I've drawn in blue a different coordinate system (u,v) (two lines that are perpendicular, or orthogonal, which means that movement along the "u" direction doesn't change your position in the "v" direction). Note that the hypotenuse is approximately 2.0492 in the "u" direction, and 4.5608 in the "v" direction. And to the degree that your calculator can show, 2.0492² + 4.5608² = 25. That is because the way we measure distance is the same as the equation for a circle: x²+y²=r², but (x,y) are just as "valid" to choose as (u,v) so u²+v²=r² has to hold as well.

A right triangle is simply the measuring of the hypotenuse with respect to some basis (be it x,y or u,v or whatever). It's the right angle (orthogonal) relationship that allows the sides to be the basis for the hypotenuse. And the Pythagorean Theorem is the direct result.