Which formula expresses the distance between two points in a Cartesian plane?

The formula that expresses the distance between two points in a Cartesian plane is derived from the Pythagorean theorem. When you have two points, ((x_1, y_1)) and ((x_2, y_2)), the distance (d) between these points is found by considering the horizontal and vertical distances between the two points.

According to the theorem, if we treat the difference in the x-coordinates, ((x_2 - x_1)), as one leg of a right triangle and the difference in the y-coordinates, ((y_2 - y_1)), as the other leg, the distance can be calculated as the hypotenuse of that triangle. Therefore, to find the distance, the formula is:

[

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

]

This shows that you square the differences in both coordinates, sum them, and then take the square root of that sum. The formula correctly incorporates both the horizontal and vertical components of distance, ensuring that we account for the overall distance between any two points in a two-dimensional space.

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