What is the distance formula in a coordinate plane?

The distance formula in a coordinate plane is a mathematical expression used to determine the distance between two points, given their coordinates ((x_1, y_1)) and ((x_2, y_2)). The formula is derived from the Pythagorean theorem. When you consider a right triangle formed by these two points, the horizontal leg of the triangle represents the difference in the x-coordinates ((x_2 - x_1)), while the vertical leg represents the difference in the y-coordinates ((y_2 - y_1)).

To find the distance between the two points, you apply the Pythagorean theorem:

1. Square the lengths of the legs: ((x_2 - x_1)^2) for the horizontal leg and ((y_2 - y_1)^2) for the vertical leg.

2. Add these two squares together: ((x_2 - x_1)^2 + (y_2 - y_1)^2).

3. Take the square root of this sum to find the length of the hypotenuse, which represents the distance (d): (d = \sqrt{(x_2