# Pythagorean theorem

In mathematics, the ** Pythagorean theorem** or **Pythagoras's theorem** is a statement about the sides of a right triangle.

One of the angles of a right triangle is always equal to 90 degrees. This angle is the right angle. The two sides next to the right angle are called the legs and the other side is called the hypotenuse. The hypotenuse is the side opposite to the right angle, and it is always the longest side.

The Pythagorean theorem says that the area of a square on the hypotenuse is equal to the sum of the areas of the squares on the legs. In this picture, the area of the blue square added to the area of the red square makes the area of the purple square. It was named after the Greek mathematician Pythagoras:

If the lengths of the legs are *a* and *b*, and the length of the hypotenuse is *c*, then, **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a^2+b^2=c^2}**
.

There are many different proofs of this theorem. They fall into four categories:

- Those based on linear relations: the algebraic proofs.
- Those based upon comparison of areas: the geometric proofs.
- Those based upon the vector operation.
- Those based on mass and velocity: the dynamic proofs.<ref>Loomis, Elisha S. 1927.
*The Pythagorean proposition: its proofs analysed and classified and bibliography of sources*. Cleveland, Ohio.</ref>

## Proof

One proof of the Pythagorean theorem was found by a Greek mathematician, Eudoxus of Cnidus.

The proof uses three lemma (mathematics):

- Triangles with the same base and height have the same area.
- A triangle which has the same base and height as a side of a square has the same area as a half of the square.
- Triangles with two congruent sides and one congruent angle are congruent and have the same area.

The proof is:

- The Template:Fontcolor triangle has the same area as the Template:Fontcolor triangle, because it has the same base and height (lemma 1).
- Template:Fontcolor and Template:Fontcolor triangles both have two sides equal to sides of the same squares, and an angle equal to a straight angle (an angle of 90 degrees) plus an angle of a triangle, so they are congruent and have the same area (lemma 3).
- Template:Fontcolor and Template:Fontcolor triangles' areas are equal because they have the same heights and bases (lemma 1).
- Template:Fontcolor triangle's area equals area of Template:Fontcolor triangle's area, because

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle {\color{blue}A_{blue}}={\color{green}A_{green}}={\color{red}A_{red}}={\color{yellow}A_{yellow}} }**

- The Template:Fontcolor triangles have the same area for the same reasons.
- Template:Fontcolor and Template:Fontcolor each have a half of the area of a smaller square. The sum of their areas equals half of the area of the bigger square. Because of this, halves of the areas of small squares are the same as a half of the area of the bigger square, so their area is the same as the area of the bigger square.

### Proof using similar triangles

We can get another proof of the Pythagorean theorem by using similar triangles.

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \frac{d}{a} = \frac{a}{c} \quad \Rightarrow \quad d = \frac{a^2}{c}\quad (1)}**

From the image, we know that **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle c = d + e \,\! }**
. And by replacing equations (1) and (2):

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle c = \frac{a^2}{c} + \frac{b^2}{c} }**

Multiplying by c:

**Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle c^2 = a^2 + b^2 \,\!.}**

## Pythagorean Triples

Pythagorean Triples or Triplets are three whole numbers which fit the equation **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle a^2+b^2=c^2}**
.

The triangle with sides of 3, 4, and 5 is a well known example. If a=3 and b=4, then **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle 3^2+4^2=5^2}**
because **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle 9+16=25}**
. This can also be shown as **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \sqrt{3^2+4^2}=5.}**

The three-four-five triangle works for all multiples of 3, 4, and 5. In other words, numbers such as 6, 8, 10 or 30, 40 and 50 are also Pythagorean triples. Another example of a triple is the 12-5-13 triangle, because **Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "http://tcs.nju.edu.cn:7231/localhost/v1/":): {\displaystyle \sqrt{12^2+5^2}=13}**