Midpoint

The point exactly between two others is just the average of their coordinates.

G0.2 already built the midpoint of a segment, geometrically. This lesson reads the same point off in numbers.

Two points, A and B, joined by a segment, with their midpoint M marked on it. A live reading gives A and B’s coordinates, and M’s coordinates as the average of each.

A labelled x and y axis with two draggable points, A and B, joined by a segment, and their midpoint M marked on it. A live reading gives both points' coordinates and M's coordinates as the average of each. This needs JavaScript switched on.

Drag A or B. M always lands exactly halfway, in both coordinates at once.

Drag A or B until they sit on opposite sides of the y-axis — the averaging still works even when one coordinate is negative and the other positive.

The point exactly halfway between two others, in both coordinates at once, is their midpoint — the same word G0.2 already used, now given a formula.

Averaging twice, not once

Halfway between two numbers on a line is their average — that is what “halfway” means for numbers. A midpoint is halfway in the x-direction and halfway in the y-direction, at the same time, so its coordinates are just that same averaging, done twice:

M = \left( \frac{x_A + x_B}{2}, \frac{y_A + y_B}{2} \right)

Most shapes are described by their corners, and a great deal of what a shape’s construction needs — centres of symmetry, the point a diagonal is bisected at, the anchor for a scaled copy — turns out to be a midpoint of two of those corners. Computing it directly from coordinates, with no drawing required, is what makes those other facts checkable by arithmetic instead of by eye.

One to watch out for.

diagonals-bisect showed a parallelogram’s diagonals cross at their shared midpoint. With coordinates, that claim becomes checkable directly: compute the midpoint of each diagonal separately, and see that both calculations land on the same pair of numbers.

Trying it on numbers

Find the midpoint of $(0, 0)$ and $(6, 8)$ .

$\left( \frac{0+6}{2}, \frac{0+8}{2} \right) = (3, 4)$ .

Find the midpoint of $(1, 5)$ and $(4, 1)$ .

$\left( \frac{1+4}{2}, \frac{5+1}{2} \right) = (2.5, 3)$ .

A segment has one endpoint at $(-2, 3)$ and midpoint at $(1, 1)$ . What is the other endpoint?

If $M$ is the average of $A$ and $B$ , then $B = 2M - A$ : $x = 2(1) - (-2) = 4$ , $y = 2(1) - 3 = -1$ . The other endpoint is $(4, -1)$ .

Do the exercise

This lesson counts as done once you get its exercise right. Nothing to tick off by hand.