Mathématiques

Midpoint of a Line Segment: Average the Coordinates

The midpoint is the average of the two endpoints' coordinates. Worked on (2, 3) and (8, 11) — and reversed, to find an endpoint from the midpoint.

Midpoint

The midpoint of a line segment is the point exactly halfway between its two endpoints, found by averaging their x-coordinates and their y-coordinates.

Où les élèves le rencontrent: Grade 7 mathematics in coordinate geometry, and again at GCSE when a perpendicular bisector or the centre of a circle has to be located.

La réponse en bref
The midpoint of a line segment is the point exactly halfway along it. Find it by averaging the two x-coordinates and averaging the two y-coordinates: the midpoint of (2, 3) and (8, 11) is (5, 7). The same formula run backwards recovers an endpoint when the midpoint is known.

Un exemple

Midpoint of (2, 3) and (8, 11) = ((2+8)/2, (3+11)/2) = (5, 7)

Average the x-values: 2 and 8 give 5. Average the y-values: 3 and 11 give 7. The midpoint is (5, 7). A quick sanity check is that both coordinates must land between the two given ones — 5 sits between 2 and 8, and 7 between 3 and 11. If either fails that test, the arithmetic has gone wrong.

Average the coordinates, one axis at a time

Halfway across and halfway up. Handle the x-coordinates on their own, then the y-coordinates on their own, and the two averages together give the midpoint. There is nothing to remember beyond that, which is why the formula written as ((x₁+x₂)/2, (y₁+y₂)/2) tends to look harder than the idea it encodes.

The method is unbothered by negatives and by decimals. The midpoint of (−4, 6) and (3, −1) is (−0.5, 2.5), and there is no rule saying a midpoint must have whole-number coordinates — half of an odd total is meant to end in .5.

It also extends without changing shape. In three dimensions the midpoint of (x₁, y₁, z₁) and (x₂, y₂, z₂) is found by averaging all three pairs, one axis at a time, exactly as in two dimensions.

The question in reverse

Exams often give one endpoint and the midpoint and ask for the other endpoint. Reaching for the formula and solving an equation works, but the quicker route is to think of the journey. From A(2, 3) to M(5, 7) the step is 3 across and 4 up. Repeating that step from M lands on (8, 11), which is B.

The algebraic version says the same thing: double the midpoint and subtract the known endpoint. That gives (2×5 − 2, 2×7 − 3) = (8, 11). Use whichever you trust, but the stepping method makes it obvious when an answer is impossible, because the midpoint must always end up between the two ends.

What the midpoint is used for later

A perpendicular bisector needs two things: the midpoint of a segment, and a gradient that is the negative reciprocal of the segment's. The midpoint supplies the point the line has to pass through.

It also settles questions about shapes. If the diagonals of a quadrilateral share the same midpoint, the diagonals bisect each other and the shape is a parallelogram. And given the two ends of a diameter, the midpoint is the centre of the circle — which is how a circle equation is recovered from two points on it.

Questions fréquentes

What is the midpoint formula?

For endpoints (x₁, y₁) and (x₂, y₂), the midpoint is ((x₁+x₂)/2, (y₁+y₂)/2). In words: average the x-coordinates, then average the y-coordinates. It is an average rather than a subtraction, which is what separates it from the distance and gradient formulas that use the same two points.

How do you find the other endpoint if you know the midpoint?

Double the midpoint's coordinates and subtract the known endpoint's. With A(2, 3) and midpoint (5, 7), the other end is (10 − 2, 14 − 3) = (8, 11). The same result comes from noticing the step from A to the midpoint and taking that step once more.

Can a midpoint have negative or decimal coordinates?

Yes to both. The midpoint of (−4, 6) and (3, −1) is (−0.5, 2.5). Nothing requires it to sit on a grid intersection, and an answer ending in .5 is usually a sign the arithmetic is right rather than wrong, since averaging two numbers of different parity always produces a half.

Is the midpoint formula related to the distance formula?

They use the same two points but ask different questions. The midpoint averages the coordinates and returns a point; the distance formula subtracts them and applies Pythagoras to return a length. Confusing the two shows up immediately, because a distance is a single number and a midpoint is a pair.

Dernière mise à jour 21 août 2026