Which formula represents the distance between two points in a coordinate plane?

The formula that represents the distance between two points in a coordinate plane is derived from the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides.

When considering two points, (x1, y1) and (x2, y2), the horizontal distance (Δx) between them is represented by (x2 - x1), and the vertical distance (Δy) is represented by (y2 - y1). According to the Pythagorean theorem, the distance ( d ) between these two points can be formulated as:

[ d^2 = \Delta x^2 + \Delta y^2 ]

This indicates that ( d^2 ) (the square of the distance) is equal to the square of the change in the x-coordinates plus the square of the change in the y-coordinates. To find the actual distance, you would take the square root of the entire equation, but the expression for distance squared accurately articulates how the distances in both dimensions contribute to the total distance between the two points, which is why this formula is considered