Midpoint is one of 3 coordinates formulas in the geometry section of this library, and it is used at high school level.
Why midpoint works
The average of two numbers sits exactly halfway between them, because it is the same amount above the smaller one as it is below the larger one. Doing that separately to the horizontal coordinates and to the vertical coordinates lands you halfway across and halfway up, which is halfway along the segment.
What each symbol means
The endpoint coordinates are averaged component by component.
Midpoint: when it holds
Both points must be expressed in the same coordinate system.
When it stops applying
It locates the middle of a straight segment only. The halfway point along a curved path, such as an arc of a circle, is not the average of the two endpoints, because that average falls inside the circle rather than on the arc.
Midpoint: a worked example
The midpoint of $(-2,3)$ and $(6,7)$ is $(2,5)$.
The mistake to avoid
What people do: Subtracting the coordinates before halving instead of adding them.
Why it goes wrong: Subtraction measures the gap between the points rather than naming a location, and for (−2,3) and (6,7) it produces (4,2), which does not even lie on the segment.
Do this instead: Add the two horizontal coordinates and halve the total, then do the same for the two vertical coordinates.
Midpoint: step by step
- Name the unknown, and the unit the answer has to come out in.
- Match the symbols to your values. The endpoint coordinates are averaged component by component.
- Check the conditions before substituting. Both points must be expressed in the same coordinate system.
- Substitute, keep exact values to the last line, then test the sign, size, and unit against a rough estimate — the check that catches most geometry slips.
Where this formula fits
- Subject
- Geometry formulas — 30 entries in this library
- Topic
- Coordinates
- Level
- High school
Formulas are easiest to keep when they sit inside a method rather than on a list. Use the links below to see where midpoint comes from, to check a calculation against a tool, and to practise it until you can recall it without looking.
- Equation of a Circle — the lesson behind this formula: read and build circle equations on the coordinate plane.
- Geometry Calculator — check your substitution and the value it produces.
- Study geometry — the subject guide that explains the ideas these formulas compress.
- Geometry Practice — questions that make you retrieve the formula instead of recognising it.
- All 30 geometry formulas — the full grouped reference, or the complete formula library.
Questions about midpoint
Does it work in three dimensions?
Yes. Average the third coordinate as well, and the same averaging idea keeps working for any number of coordinates you care to add.
Why is there a plus sign when the distance formula uses a minus?
Averaging needs a total, so the coordinates are added. Measuring a separation needs a gap, so there they are subtracted instead.
How do I find an endpoint from the midpoint and the other end?
Double each coordinate of the midpoint and subtract the matching coordinate of the known endpoint, since the midpoint is their average.
Can I use it to find a point one third of the way along?
Not directly. Weight the endpoints instead, taking two thirds of the starting point plus one third of the finishing point.