Endpoint Math Formula

Students learn how to apply the endpoint math formula — a derivation of the midpoint formula — during a unit on graphing in the coordinate plane, which is typically taught in an algebra course but sometimes covered in a geometry course. To use the endpoint math formula, you must already know how to solve two-step algebraic equations.

Problem Setup

Problems involving the endpoint math formula involve three points of a line segment: the two endpoints and the midpoint. You are given the midpoint and one endpoint and asked to find the other endpoint. The formula to use is a derivation of the better-known midpoint formula. Letting (m1, m2) represent the given midpoint, (x1, y1) represent the given endpoint, and (x2, y2) represent the unknown endpoint, the formula is: (x2, y2) = (2_m1 – x1, 2_m2 – y1).

Worked Example

Suppose you are given a midpoint of (1, 0), one endpoint of (-2, 3) and asked to find the other endpoint. In this example, m1 = 1, m2 = 0, x1 = -2, y1 = 3 and x2 and y2 are the unknowns. Substituting the known values into the aforementioned formula produces (x2, y2) = (2_1 – -2, 2_0 – 3). Simplify using the order of operations — that is, first perform the multiplication, and then perform the subtraction. Doing so yields (x2, y2) = (2 – -2, 0 – 3), which then becomes (x2, y2) = (2+2, 0 – 3), resulting in a final answer of (x2, y2) = (4, -3). If you wish, you may check your solution by substituting all points into the midpoint formula: (m1, m2) = {[(x1 + x2)/2], [(y1 + y2)/2]}.