What is the formula used to calculate the distance between two points (X1, Y1) and (X2, Y2)?

The correct choice for calculating the distance between two points ((X_1, Y_1)) and ((X_2, Y_2)) is derived from the Pythagorean theorem. The formula for distance is expressed as:

[ d = \sqrt{(X_2 - X_1)^2 + (Y_2 - Y_1)^2}

]

This formula reflects the relationship between the horizontal and vertical distances (the legs of a right triangle) formed by these two points on a Cartesian plane. By taking the difference between the x-coordinates and the y-coordinates, you can find the lengths of these legs, which, squared and summed, gives you the square of the length of the hypotenuse (the distance between the points).

To clarify why other options do not represent the correct formula:

The first option simply provides the squared values of the differences without taking the square root, which does not yield the actual distance but rather the squared distance.

In the second option, while it has the correct structure of the distance formula, it incorrectly positions the square root over the expression by including ( (X_2 - X_1) ) without being squared, leading to an inaccurate calculation