Distance Formula

Start by plotting A(x_1,y_1) and B(x_2,y_2) on the coordinate plane. Both points can be arbitrarily plotted in Quadrant I for simplicity. Note that the position of the points in the plane does not affect the proof. Assume that x_2 is greater than x_1 and that y_2 is greater than y_1.

Next, draw a right triangle. The hypotenuse of this triangle will be the segment that connects points A and B.

The difference between the x-coordinates of the points is the length of one of the legs of the triangle. Furthermore, the length of the other leg is given by the difference between the y-coordinates. Therefore, the lengths of the legs are x_2-x_1 and y_2-y_1. Now, consider the Pythagorean Equation. a^2+b^2=c^2 Here, a and b are the lengths of the legs, and c the length of the hypotenuse of a right triangle. Substitute the expressions for the legs for a and b to find the hypotenuse's length. Then, the equation can be solved for c.

Note that, when solving for c, only the principal root was considered. The reason is that c represents the length of a side and therefore must be positive. Keeping in mind that c is the distance between A(x_1,y_1) and B(x_2,y_2), then c=d. By the Transitive Property of Equality, the Distance Formula is obtained. c= sqrt((x_2-x_1)^2+(y_2-y_1)^2) c= d ⇓ d= sqrt((x_2-x_1)^2+(y_2-y_1)^2)