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# What Is the Pythagorean Theorem? What is the Pythagorean Theorem? Why is it so useful? And who came up with the idea in the first place? Keep on reading to find out!

By
Jason Marshall, PhD
Episode #155 Almost everybody has heard of the Pythagorean Theorem. Heck, even the brainless Scarecrow from the Wizard of Oz knew about it. Remember? When he receives his diploma from the “Wizard” at the end, he declares: “The sum of the square roots of any two sides of an isosceles triangle is equal to the square root of the remaining side.”

Those of you who are already familiar with this famous theorem may have noticed a little problem here since the Scarecrow—or maybe the screenwriters—got it wrong! Perhaps it was just that he didn’t receive a brain after all? What should the Scarecrow have said? By which I mean what does the Pythagorean theorem really say? That’s exactly the question we’ll be answering today.

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## The Pythagorean Theorem

In words, the Pythagorean Theorem says that the lengths of the two legs of a right triangle—which is a triangle in which two sides come together to form a right angle—have a very special relationship to the length of its long side…aka, its hypotenuse. Specifically, if you square the lengths of the two legs and add the resulting numbers together, that number will always equal the square of the length of the hypotenuse.

Words are nice, but we’ve been talking about algebra lately, so let’s see what the Pythagorean Theorem looks like in terms of variables a, b, and c. Namely:

a^2 + b^2 = c^2

What do the symbols a, b, and c here mean? Well, a and b represent the lengths of the shorter legs of the triangle, and c represents the length of the hypotenuse. If you think about it for a minute, you’ll see that this formula “says” the exact same thing as the more wordy description from before…but it’s a lot more concise.

## How to Use the Pythagorean Theorem

The beauty of this algebraic form of the Pythagorean Theorem is that I can tell you the lengths of the legs of any right triangle, a and b, and you can tell me how long the hypotenuse, c, has to be. Just to make sure this is true, let’s check and see if it works for the ingenious rope that Knot Dude—the hero of last week’s episode—used to help his father build the Great Pyramid of Giza.

As you’ll recall, Knot Dude’s rope triangle had a=3 and b=4. Since a^2=9, b^2=16, and a^2 + b^2 is therefore equal to 9 + 16 = 25, the Pythagorean Theorem tells us that c^2—the square of the length of Knot Dude’s triangle’s hypotenuse—must also equal 25. Is that true? Well, the length of the hypotenuse of Knot Dude’s triangle was 5, which means that c^2 was indeed equal to 25. The Pythagorean Theorem works!

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