What is the distance formula used to calculate the distance between two points in a Cartesian plane?

The distance formula for calculating the distance between two points in a Cartesian plane is derived from the Pythagorean theorem. When you have two points, (x1, y1) and (x2, y2), you can visualize a right triangle where the horizontal leg represents the difference in x-coordinates (x2 - x1) and the vertical leg represents the difference in y-coordinates (y2 - y1).

According to the Pythagorean theorem, the length of the hypotenuse, which represents the distance (d), is calculated by taking the square root of the sum of the squares of the lengths of the two legs. Thus, the correct formula is:

d = √((x2 - x1)² + (y2 - y1)²).

This formulation ensures that the resulting distance is always a non-negative value, as the squares of the differences cannot be negative. The other alternatives do not represent this correct method of calculating distance, either omitting the square root or incorrectly structuring the terms, which could lead to incorrect results.