How can the distance formula be derived from the pythagorean theorem?

We start by placing a right triangle in the coordinate plane as seen below. Notice that we have given the vertices arbitrary x- and y-coordinates.

Notice that, although the coordinates are arbitrary, the y-coordinates of the vertices on the horizontal leg are the same since they will have the same y-value. Likewise, the x-coordinates of the vertices on the vertical leg are the same as the will have the same x-value.

The Pythagorean Theorem states that the sum of the squares of the length of the legs is equal to the square of the length of the hypotenuse.

To find the lengths of the legs in terms of their coordinates we will use the of the differences of their coordinates as seen in the figure below.

So, according to the Pythagorean Theorem

##(|x_2-x_1|)^2+(|y_2-y_1|)^2=d^2##

We can drop the absolute value symbols since we are squaring the differences in the coordinates (the result will be positive).

##(x_2-x_1)^2+(y_2-y_1)^2=d^2##

Taking the square root of both sides of the equation results in the .

##d=sqrt((x_2-x_1)^2+(y_2-y_1)^2)##