What formula is used to calculate the distance between two coordinates (x₁, y₁) and (x₂, y₂)?

The formula that is used to calculate the distance between two points in a Cartesian coordinate system, given as (x₁, y₁) and (x₂, y₂), is derived from the Pythagorean theorem. According to this theorem, the distance (d) can be interpreted as the hypotenuse of a right triangle formed by the horizontal and vertical differences between the points.

The horizontal distance is represented by (x₂ - x₁), and the vertical distance is represented by (y₂ - y₁). Therefore, to find the overall distance, you square each of these differences, sum them, and then take the square root of the result. This results in the formula: d = √[(x₂ - x₁)² + (y₂ - y₁)²].

This formula effectively measures the straight-line distance between the two points in the 2D space, which is particularly important in various applications of mathematics, physics, and engineering. The geometric interpretation of the formula as it relates to the right triangle reinforces its validity in calculating distance accurately.