How can you picture the meaning of the Pythagorean Theorem? What are the geometric meanings of expressions, equations, and all of algebra? Keep on reading to find out!
As we’ve seen, an equation like the Pythagorean Theorem is really a kind of rule saying that whenever we plug in numbers for the variables in the equation, the combination of values in the expression on the left side of the equals sign must have the same value as the combination in the expression on the right.
At least, that’s the algebraic way of looking at things. And while this algebraic viewpoint is helpful, fantastic, and all-around super important, it’s good to keep in mind that it’s not the only way to think about equations. To show you what I mean, today we’re going to learn how to look at math from another angle. In particular, we’re going to learn how to visualize the meaning of everybody’s favorite theorem—the Pythagorean Theorem—using just a simple picture and some clever thinking.
A Picture of the Pythagorean Theorem
To get things started, I want you to picture one of those beautiful 3–4–5 triangles that we’ve been talking about lately while uncovering the long lost story of Knot Dude. Remember, the 3–4–5 here means that the two legs of the right triangle are 3 and 4 units long, and that the hypotenuse, c, is 5 units long.
Next, draw perfect squares attached to each side of your triangle. The lengths of the sides of the squares should be the same as the lengths of the side of the triangle they’re attached to. In other words, the side of the triangle with a length of 3 should end up attached to a square whose sides are 3 units long, and so on. Finally, further divide up each of these squares you’ve drawn into a bunch of little 1-by-1 unit squares.
Why in the world did we do this? In particular, what’s up with all of those 1-by-1 unit squares? Believe it or not, those little guys are going to play a key role in helping you picture the meaning of the Pythagorean Theorem. How? Before I tell you, I encourage you to first see if you can figure it out first. Then, when you’re ready, continue on!