How is the Pythagorean theorem used to measure distance in coordinate space?

In Euclidean geometry, the Pythagorean theorem serves as the foundation for calculating the Euclidean distance between two points in a coordinate system.

By identifying two points, such as $(a,b)$ and $(c,d)$ in a two-dimensional plane, one can construct a right triangle by introducing a third point at $(c,b)$. The horizontal and vertical legs of this triangle represent the differences between the coordinates, $(a-c)$ and $(b-d)$. According to the Pythagorean theorem, the square of the hypotenuse—which represents the straight-line distance between the original two points—is equal to the sum of the squares of these legs. This results in the distance formula:

$$\sqrt{(a-c{)}^2+(b-d{)}^2}$$

This principle extends to three-dimensional space, where the distance between points $(a,b,c)$ and $(d,e,f)$ is calculated as:

$$\sqrt{(a-d{)}^2+(b-e{)}^2+(c-f{)}^2}$$

These formulas provide a precise method for determining the length of a straight line segment between any two points in Euclidean space.12