What is the distance formula between two points (x1, y1) and (x2, y2)?

The distance formula calculates the length of the segment connecting two points in a two-dimensional space. The correct formula is derived from the Pythagorean theorem, which relates the sides of a right triangle to its hypotenuse.

For two points ((x_1, y_1)) and ((x_2, y_2)), you can think of these points as forming a right triangle, where the horizontal distance (the difference in x-coordinates) is one leg and the vertical distance (the difference in y-coordinates) is the other leg. The lengths of these legs of the triangle are given by ((x_2 - x_1)) and ((y_2 - y_1)). According to the Pythagorean theorem, the square of the length of the hypotenuse (the distance between the two points) is equal to the sum of the squares of the lengths of the other two sides.

Thus, the distance (d) between the points is calculated as:

[

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

]

This formulation directly corresponds to the correct choice, encapsulating