Which formula is used to find the length of a segment between two points?

The distance formula is used to find the length of a segment between two points in a coordinate plane. This formula is derived from the Pythagorean theorem and calculates the straight-line distance between two points with coordinates ((x_1, y_1)) and ((x_2, y_2)). It is expressed as:

[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

]

In this formula, the difference in the x-coordinates and the y-coordinates of the two points is squared, summed, and then the square root of that sum gives the distance. This is crucial in geometry for understanding spatial relationships and calculating lengths, making it essential for various applications.

The area formula, perimeter formula, and volume formula serve different purposes in geometry. The area formula calculates the space enclosed within a shape, the perimeter formula adds up the lengths of all sides of a shape, and the volume formula determines the space within a three-dimensional object. Each of these is distinct from finding the length of a segment directly between two points. Hence, the distance formula is the appropriate choice for this task.