The Pythagorean Theorem

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The Pythagorean theorem deals with the lengths of the sides of a right triangle.

It is often written in the form of the equation:

a2 + b2 = c2

The theorem states that:

The sum of the squares of the lengths of the legs of a right triangle ('a' and 'b' in the triangle shown below) is equal to the square of the length of the hypotenuse ('c').

1. Pythagorean Puzzle - The Proof - Nova (PBS)
2. Pythagorean Theorem - Dr. Math Archives

When would I use the Pythagorean theorem?

The Pythagorean theorem is used any time we have a right triangle, we know the length of two sides, and we want to find the third side. For example:

I was in the furniture store the other day and saw a nice entertainment center on sale at a good price. The space for the TV set measured 17" x 21". I didn't want to take the time to go home to measure my TV set, or get the cabinet home only to find that it was too small.

I knew my TV set had a 27" screen, and TV screens are measured on the diagonal. To figure out whether my TV would fit, I calculated the diagonal of the TV space in the entertainment center using the Pythagorean theorem:

172 + 212 = 289 + 441 = 730

So the diagonal of the entertainment center is the square root of 730, which is about 27.02".

Sounds like my TV should fit, but the 27" diagonal on the TV set measures the screen only, not the housing, speakers and control buttons. These extend the TV set's diagonal several inches, so I figured that my TV would not fit in the cabinet. When I got home, I measured my TV set and found that the entire set was 21" x 27.5", so it was a good decision not to buy the entertainment center.

The Pythagorean theorem is also frequently used in more advanced math. The applications that use the Pythagorean theorem include computing the distance between points on a plane; converting between polar and rectangular coordinates; computing perimeters, surface areas and volumes of various geometric shapes; and calculating maxima and minima of perimeters, or surface areas and volumes of various geometric shapes.

One of the most common applications of the Pythagorean theorem is in the distance formula. To find the distance between between two points, the distance formula states:

The in the formula stands for "difference between," so
x = x2 - x1  and  y = y2 - y1.

To see how this uses the Pythagorean theorem, square both sides. Then we have:

d2 = x2 + y2

Can you figure out the right triangle that this describes?

1. Using the Pythagorean Theorem - Nova (PBS)
2. Pole in a Box - Dr. Math Archives
3. Ratios and Geometry - Dr. Math Archives
4. About Rational Numbers - Dr. Math Archives

How can we prove the Pythagorean theorem is right?

There are several different ways of proving the Pythagorean theorem. Here's one way:

Let's start by looking at a square whose side length is (a+b). We can mark a point on the side that divides it into segments of length a and b. Here are three examples, using different lengths for legs a and b:

Inside the blue square let's construct a yellow square of sidelength c. Its corners must touch the sides of the blue square. The remainder of the space will consist of four blue congruent abc triangles. Here it is for our example squares:

In each case, the area of the larger blue square is equal to the sum of the areas of the blue triangles and the area of the yellow square.

Since the area of a square is (sidelength)2 and the area of a triangle is 1/2(base)(height), we can write the equation:

```          (a+b)2 = c2 + 4[(1/2)ab]
```

Simplifying:

```          (a+b)2 = c2 + 4[(1/2)ab]
(a+b)(a+b) = c2 + 2ab
a2 + 2ab + b2 = c2 + 2ab
```

Now subtract the 2ab from both sides of the equation, and we have the Pythagorean theorem:

```         a2 + b2 = c2

```

1. Euclid's Elements - Book I, Proposition 47 - David Joyce
2. A Proof of the Pythagorean Theorem - Isaac Reed
3. Pythagorean Theorem - Paul Flavin
4. Proofs of The Pythagorean Theorem - Susan Addington
5. Pythagorean Theorem - Interactive Mathematics Miscellany and Puzzles - Bogomolny
6. Proving the Pythagorean Theorem - Dr. Math Archives
7. Pythagorean Theorem: Why Use the Converse? - Dr. Math Archives

For many more links, search the Math Forum site for the exact phrase: Pythagorean theorem. And browse related problems in the Geometry Problem of the Week archive.