The Pythagorean Theorem

The Pythagorean Theorem is one of the earliest laws that we are taught in school, and ends up relating to both Trigonometry . The Theorem allows us to find the third side of a right triangle as long as we know the other two sides.

The History of the Pythagorean Theorem

This is not nesecary to learn the mathematics, but is quite interesting

The Pythagorean Theorem is named for the Greek Philosipher and Mathematician, Pythagoras of Samos. He lived sometime between 569 and 475 BC. Pythagoras (Pronounced “Pie-Thag-O-Russ”, not “Pie-That-O-Rass”), though, did not truly invent the Pythagorean Theorem, but made it possible use it algebraically as opposed to using Pythagorean Triples . Before him, many various mathematicians, including Chineese, Arabics, and Egyptians, invented various proofs and lists of triples that used the same concept, but without the algebra. These men and women used their own “Pythagorean Theorems” and Triples to help do things like build canals and pyramids.

What Is the Pythagorean Theorem?

The Pythagorean Theorem states that there is a consatnt relationship between the lengths of the sides and the hypotenuse in a right triangle. To put it simply, a2+b2=c2,where a and b are the lengths of the legs and c is the length of the hypotenuse (the longest side, opposite the right angle). We can, using that formula, find any 3rd side if we know the other two.

Using the Pythagorean Theorem

Lets say that we need to find the length of one of the legs of our triangle.

Pythagorean Theorem

That’s our triangle, with sides a b and c. Let us propose that the length of side a is 3 and the length of side c is 5. How can we find the length of side b? We can do this quite easily using the pythagorean theorem.

Of course, the first step is to set up our equation, substituting the known values in for the variables.



Next, we need to figure out 32 and 52. These are of course 9 and 25, respectively.


Now, we just follow our standard rules for equations- get the varialbes on one side, numbers on the other

Subtract 9 from each side.

Now we just find the square root of each side


And there you have it, the length of side b is 4.

Now, you’re probably thinking…

Next time, why would I do all that math? I can just go and see that if theres a 3 for a leg and a 5 for the hypotenuse, the other leg is 4! Or if there is a 3 for one leg and a 4 for the other, the hypotenuse is 5! These are what we call Pythagorean Triples. There are many of them, and it pays to know them.

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