The distance between two points

The distance between two points in the coordinate plane comes from the Pythagorean theorem. Here we find the distance between $A(1, 2)$ and $B(4, 6)$.

Going from $A$ to $B$ moves $4 - 1 = 3$ across and $6 - 2 = 4$ up. Taking these horizontal and vertical differences as the two legs of a right triangle, the segment joining $A$ and $B$ is its hypotenuse. By the Pythagorean theorem,

so the distance is $5$.

In general, the distance between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given, using the horizontal difference $x_2 - x_1$ and the vertical difference $y_2 - y_1$, by the following.

Because the differences are squared, it does not matter if $x_2 - x_1$ is negative. The large dots on the graph are $A(1, 2)$ and $B(4, 6)$, and the length of the segment joining them is the distance $5$ we found.