Pythagorean theorem

In mathematics, the Pythagorean theorem or Pythagoras' 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, a² + b² = c².

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

1. Those based on linear relations: the algebraic proofs.
2. Those based upon comparison of areas: the geometric proofs.
3. Those based upon the vector operation.
4. Those based on mass and velocity: the dynamic proofs.

For the proof by Eudoxus of Cnidus, see: Pythagorean theorem/proof.

Proof using similar triangles

$\frac{d}{a} = \frac{a}{c} \quad \Rightarrow \quad d = \frac{a^2}{c}\quad (1)$

$\frac{e}{b} = \frac{b}{c} \quad \Rightarrow \quad e = \frac{b^2}{c}\quad (2)$

From the image $c = d + e \,\!$. And by replacing equations (1) and (2):

$c = \frac{a^2}{c} + \frac{b^2}{c}$

Multiplying by c:

$c^2 = a^2 + b^2 \,\!.$

Pythagorean Triples

Pythagorean Triples are three whole numbers which meet the equation a² + b² = c².

One well known example is the 3-4-5 triangle: if a=3 and b=4, 3² + 4² = 5² or 9 + 16 = 25. This can also be shown as $\sqrt{3^2+4^2}=5.$

The three-four-five triangle works for any multiples of 3, 4, and 5; such as 6, 8, and 10; or 30, 40 and 50. Another, less-often used example is the 12-5-13 triangle. $\sqrt{12^2+5^2}=13$