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Squares and Square Roots

Squares and Square Roots

At a Glance - The Pythagorean Theorem

Before we state the Pythagorean Theorem, we'll introduce a right triangle. Reader, right triangle. Right triangle, reader. We have a feeling you two will get along. You have a lot in common. You both have two legs, are most comfortable at 90 degrees, and are always playing some angle. You also have the same taste in music.

A right triangle is a triangle with a right angle in it, as indicated in the image below by the little box in the lower right corner.

The sides of the triangle that form the right angle are called legs, and the long side across from the right angle is the hypotenuse.

We really, really, really hope you like right triangles. We want them to live up to all the hypotenuse.

The Pythagorean Theorem states that, for a right triangle with legs of length a and b and a hypotenuse of length c, the following equation is true:

a2 + b2 = c2

After that dizzying quadratic formula, this one isn't bad at all. We can handle a square on each variable. That sounds okay. However, what does that mean in relation to the right triangle? In words, the sum of the squares of the lengths of the legs equals the square of the length of the hypotenuse. It's a simple concept, but a little wordy, so feel free to go back and read it again. And again. It's okay, read it a few dozen times. You'll be using it a lot.

Why is the Pythagorean Theorem true? Because Shmoop says it is? Nah, we'll back up our claims with some proof...this time. Here's one way to see why the Theorem makes sense. Take the right triangle shown above, make 3 copies of it, and lay them out like this:

Now we have a big square with side length (a + b), and a smaller square inside with side length c. Who knew triangles could make such great squares? You should see their trapezoids.

We can find the area of the big square in two ways. We know, we know: why can't there ever be only one way?

We can find the area of the big square using its side length, or we can add the areas of the four triangles and the smaller square. With the first way, we have:

With the second way, we have:

Since a2 + 2ab + b2 and 2ab + c2 are each the area of the big square, they must be equal:

a2 + 2ab + b2 = 2ab + c2

Subtracting 2ab from each side, we see that:

a2 + b2 = c2

Which is the statement of the Pythagorean Theorem. If that wasn't like magic, we don't know what is. Top that, David Blaine.

For more proofs of the Pythagorean Theorem, check this out.

Even if you don't read through them, it's sort of fun to check out how many proofs there are...especially if you find it fun to see how obsessively deranged mathematicians can be. Did we legitimately need that many proofs, guys? We believed you the first time.

We use the Pythagorean Theorem to find out how long different sides of a triangle are. Usually we'll be told the lengths of two sides of the triangle, and then be asked to find the length of the third side. Although we don't know why whoever measured those first two sides couldn't have measured the third one while he was at it. It would have been a nice gesture.

Sample Problem

A right triangle has one leg that's 3 cm long and another leg that's 4 cm long. How long is its hypotenuse?

We use the Pythagorean Theorem to figure this out. Let's call the hypotenuse c. The two legs have lengths 3 and 4, so 32 + 42 = c2.

Simplifying, we see that 25 = c2.

The solutions to this equation are c = ±5, but since c is a length, we'll only take c = 5 cm. No -5 cm long triangles for us.

You see, it's not enough to arrive at an answer; you'll need to think about it logically and see if it makes sense. If it doesn't, say buh-bye and throw it into the garbage disposal. If you were ever to visit a landfill, you'd be amazed by how many triangles with negative lengths are laying around all over the place.

Example 1

A right triangle has one leg of length 5 and a hypotenuse of length 16. What is the length of the other leg of the triangle?

Example 2

A right triangle has a hypotenuse of 30 inches. Both of its legs are the same length. How long is a leg of this right triangle?

Exercise 1

Find c in the triangle shown below:

Exercise 2

Find x in the triangle shown below:

Exercise 3

How long is the hypotenuse of a right triangle with legs that have lengths 8 and 9?

Exercise 4

A right triangle has a hypotenuse with a length of 13. One leg of the triangle is 3 times longer than the other leg of the triangle. How long are the legs of the triangle?

Exercise 5

What does the Pythagorean Theorem say for a triangle with a hypotenuse of length a and legs of lengths b and c?

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