What is the Pythagorean Theorem?

The Pythagorean theorem is used to calculate the length of a side of a right triangle when the lengths of the other sides are known. It is defined as:

    equation

where equation is the hypothenuse (longest side in the right triangle), and equation and equation are the other two sides as shown in the following diagram:

righttriangle

Derivation of the Pythagorean theorem

Let us understand how this formula is derived. If we duplicate and mirror our triangle, we can put them together to get a rectangle where the hypothenuse equation is the diagonal. Let us duplicate this some more and arrange as shown in the following diagram:

triangles

We arranged the rectangles (each consisting of two triangles) such that the diagonals equation form a square which we highlight in green:

triangles2

We know that the area of the green square is equation. Our goal is to find this area in terms of the other sides equation and equation. The green square consistes of four triangles and a smaller square. Each triangle, as shown in the following diagram can be calculated as equation.

triangles4

We have four triangles in total, which gives us an area of equation. We still need to find the area of the inner square:

triangles3

We can see that the length of the side is equation, so the area of the square is equation. This means that the area of the larger square which we found to be equation can also be expressed as the sum of the four triangles and the small square equation. We can set them equal and simplify:

(1)   equation

We derived the Pythagorean theorem. We can use it to calculate the length of a side of the triangle when two other sides are known.

Example

Given a triangle with sides equation and equation, find the length of side equation:

(2)   equation