What does the distance formula calculate?

The distance formula calculates the distance between two points in a Cartesian coordinate system. The points are typically represented as (x₁, y₁) and (x₂, y₂).

The correct expression for this distance is given by d = √[(x₂ - x₁)² + (y₂ - y₁)²]. This formula is derived from the Pythagorean theorem, where the difference in the x-coordinates and the difference in the y-coordinates form the two legs of a right triangle. The distance between the two points represents the hypotenuse.

Taking the squared differences, (x₂ - x₁)² and (y₂ - y₁)², reflects how much the points differ in each coordinate direction. By summing these squared differences and taking the square root, the formula gives the straight-line distance, or the direct linear path between the two points. This method ensures that the distance is always a non-negative value, as it should represent a physical length.

Other processes, such as summing the absolute differences or simple arithmetic addition of coordinates, do not provide the actual distance between two points but rather result in other mathematical properties that do not pertain to geometric distance. Hence