Distance

Now we return to two-dimensional space, where the notion of distance is a familiar one. In the figure below, and are two points in the plane, and the distance between them is the hypotenuse of the triangle with vertices

or

by the theorem of Pythagoras.

It is of some interest to give a very simple and often unfamiliar proof of the theorem of Pythagoras, as follows: We wish to prove that if and are the legs of a right triangle and is its hypotenuse, then . Let us form a square of side , and decompose it in two ways:

From (1), the area of the square is

where is the area of a right triangle with legs a and b. From (2), its area is

Thus

or
which was to be proved.

By applying the theorem of Pythagoras twice, we see that the distance between two points and in 3-space is

Thus we are led to define the distance between two points and in n-space to be

We observe that this is equal to

We also note that the distance between two points is 0 only if the points are not actually distinct, i.e., only if . (Why?)