Consider AB shown in the coordinate plane below. The horizontal and vertical distance, Δx and Δy, respectively, between its endpoints are shown.
The midpoint of this segment, M, divides the distance between its endpoints exactly in half both horizontally and vertically. That means, the distance from each endpoint to the midpoint is half of the established distances.
Focusing first on the
x-coordinates, the horizontal distances between the midpoint and the endpoints, are given by
∣xm−x1∣and∣x2−xm∣,
respectively. Here, the absolute values are used to ensure the distances are positive. Since these distances are equal the following relationship is true.
∣xm−x1∣=∣x2−xm∣
This equation can be solved to find the
x-coordinate of the midpoint,
xm. ∣xm−x1∣=∣x2−xm∣ xm−x1=x2−xm 2xm−x1=x2 2xm=x2+x1 2xm=x1+x2 xm=2x1+x2
Thus, the
x-coordinate for the midpoint is
xm=2x1+x2. In the same way, it can be shown that the
y-coordinate for
M is
ym=2y1+y2. Therefore, the midpoint of a segment is
M(2x1+x2,2y1+y2).