What is the distance formula in a coordinate plane?

The distance formula in a coordinate plane is derived from the Pythagorean theorem, which relates the lengths of the sides of a right triangle to the length of the hypotenuse. When finding the distance between two points ((x_1, y_1)) and ((x_2, y_2)), you form a right triangle where the legs are the differences in the x-coordinates and y-coordinates: ((x_2 - x_1)) and ((y_2 - y_1)).

By applying the Pythagorean theorem, the distance (d) (which represents the length of the hypotenuse) can be calculated using the formula:

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

This correctly captures how the coordinate changes in both dimensions combine to give the straight-line distance between two points on a plane. The square root is necessary because it allows us to return to the positive length from the squares of the differences.

The other options do not accurately reflect this relationship. The addition and absolute value approaches do not take into account the need for squaring the differences to