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!
The Geometric Meaning of Equations
While it’s all well and good to know how to think about the geometric meaning of a^2, b^2, and c^2 separately, we’re really interested in knowing how we can put them all together to understand the entire Pythagorean Theorem. So, let’s do exactly what the Pythagorean Theorem says—let’s combine the big squares containing a^2 and b^2 small 1-by-1 unit boxes. In other words, let’s take all the 1-by-1 boxes that make up the 3-by-3 and 4-by-4 squares and combine them together into a bigger square.
What does this show us? Well, if you count up all the unit boxes, you’ll find that the big box created by adding a^2 1-by-1 boxes to b^2 1-by-1 boxes—for a total of a^2 + b^2 boxes—is the exact same size as the big box from the hypotenuse of the triangle that contains c^2 1-by-1 unit boxes. So what? Well, for one thing, this shows us why the Pythagorean Theorem works for our 3–4–5 triangle. And it also shows us that we now have another way to think about what the Pythagorean Theorem means. Finally, it’s a reminder that problems in algebra—and life—can usually be viewed in multiple ways, and doing so is often a great way to obtain a better understand of their true meaning.
Pythagorean Triples Puzzle
Before we finish up, I want to fill you in on the answer to last week’s brain teaser: Are there other Pythagorean triples besides the simple ones we found? And the answer is… Yes, there are an infinite number of them! In fact, if you pick any two positive integers, let’s call them x and y (where y > x), and calculate:
a = y^2 – x^2
b = 2 x y
c = x^2 + y^2
then these three numbers will always be Pythagorean triples. Go ahead and try it out with a few numbers and test if the results agree with the Pythagorean Theorem!
Okay, that’s all the math we have time for today. If you want to learn more about algebra, please check out my book The Math Dude’s Quick and Dirty Guide to Algebra.
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Until next time, this is Jason Marshall with The Math Dude’s Quick and Dirty Tips to Make Math Easier. Thanks for reading, math fans!