A. Derive the distance formula (d) shown below for points A = (x1, y1) and B = (x2, y2).

2
2 1
2
2 1 d = (x − x ) + ( y − y )
Note: Look for an application of the Pythagorean theorem where the red line, segment
AB, is the hypotenuse of a right triangle. You can then determine the lengths of the legs
and justify why they meet at right angles.

d = sqrt [ ( x2 - x1 ) ^ 2 + ( y2 - y1 ) ^ 2 ]

To derive the distance formula, we can use the Pythagorean theorem. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides.

In this case, we can consider point A (x1, y1) and point B (x2, y2) as the vertices of a right triangle. The horizontal distance between the two points can be represented by the difference in their x-coordinates (x2 - x1), and the vertical distance can be represented by the difference in their y-coordinates (y2 - y1).

If we consider the horizontal and vertical distances as the lengths of the legs of the right triangle, then the distance between the points A and B (represented by d) is the length of the hypotenuse.

Therefore, according to the Pythagorean theorem, we have:

d^2 = (x2 - x1)^2 + (y2 - y1)^2

To find the distance formula, we can take the square root of both sides:

d = √[(x2 - x1)^2 + (y2 - y1)^2]

This formula provides the distance between two points A and B in a two-dimensional plane.