# How is the pythagorean theorem related to the distance formula?

##### 1 Answer

We generally define the distance between two points using the Pythagorean theorem.

Consider a two dimensional Cartesian coordinate system (the same ideas can be extended to any number of dimensions. The natural way to think about the geometry of this coordinate system is using vectors.

The distance

where

Since the Cartesian system is orthogonal, the components of any vector in this space are orthogonal. Therefore, they form the two smaller sides of a right triangle, with the distance being the length of the hypothenuse of this triangle, wich can be seen as the vector joining

For the distances between two points

where

In more advanced Maths, it's useful to think of the distance as a more general concept, and different definitions of distance can arise, leading to the theory of metric spaces .