The midpoint of a line segment is the point exactly halfway between its two endpoints. Its coordinates are simply the averages of the endpoints’ coordinates, so it takes one addition and one division per axis. This calculator works in both directions: give it two endpoints and it returns the midpoint, or give it one endpoint and the midpoint and it finds the missing end. It handles flat (x, y) and spatial (x, y, z) coordinates, and for 2D segments it also writes the equation of the perpendicular bisector.
How to use the midpoint calculator
- Under Find the, choose Midpoint or Missing endpoint.
- Under Dimensions, choose 2D (x, y) or 3D (x, y, z).
- Enter Endpoint A as x₁, y₁ (and z₁ in 3D).
- For a midpoint, enter Endpoint B as x₂, y₂ (and z₂). For a missing endpoint, enter Midpoint M as Midpoint x, Midpoint y (and Midpoint z).
- The answer appears at the top, followed by all three points, the Length of AB and the Half length (A to M). In 2D you also get the Slope of AB, the Perpendicular bisector and a plot of the segment with its midpoint.
Midpoint formulas
Why an average? Walking from A toward B, you cover the full change Δx = x₂ − x₁. Stopping halfway puts you at x₁ + Δx ÷ 2, which rearranges to (x₁ + x₂) ÷ 2. The same holds on every axis, independently. The missing-endpoint formula is that equation solved for x₂.
The perpendicular bisector passes through M with the negative reciprocal of the segment’s slope:
If the segment is horizontal, the bisector is the vertical line x = xM; if the segment is vertical, the bisector is horizontal.
Worked example: a garden path on a site plan
Two stakes mark the ends of a straight garden path at A(10, 30) and B(70, 110), measured in feet from a corner of the lot. You want a lamp post at the middle of the path and a bench that sits equally far from both ends.
- Midpoint: ((10 + 70) ÷ 2, (30 + 110) ÷ 2) = (40, 70). Put the lamp post there.
- Path length: √(60² + 80²) = 100 ft, so the post is 50 ft from each stake.
- Slope of the path: 80 ÷ 60 = 4/3, so the bisector's slope is its negative reciprocal, −3/4.
- Perpendicular bisector: y = −(3/4)x + 100. Any bench location on this line is equidistant from both ends.
- Check: at x = 0 the line gives (0, 100), which is √5,000 ≈ 70.71 ft from both A and B.
Finding a missing endpoint
Missing-endpoint problems turn up whenever you know the center of something symmetric. If a circle is centered at (3, −1) and one end of a diameter is at (7, 2), the center is the midpoint of that diameter, so the other end is (2 × 3 − 7, 2 × (−1) − 2) = (−1, −4). The diameter is 10 units, so the radius is 5. The same calculation reflects a point through another point, as when mirroring part of a symmetric drawing or layout through its center.
A common slip is to compute M − A, which gives only the half-step from A to the midpoint. Doubling is essential: B = A + 2(M − A) = 2M − A.
Where the perpendicular bisector helps
- Circle through three points. The perpendicular bisectors of any two sides of a triangle meet at its circumcenter, the center of the circle through all three corners. Solve the two bisector equations together with the system of linear equations calculator.
- Nearest-location boundaries. The bisector between two stores, fire stations or cell towers splits the map into the area closer to each. Repeating this for many sites produces a Voronoi diagram.
- Compass-and-straightedge construction. The standard construction draws two equal circles centered on the endpoints, each with a radius longer than half the segment. The line through their two crossing points is the perpendicular bisector. The very first proposition of Euclid’s Elements starts from the same pair of circles.
To measure the segment itself use the distance between two points calculator, and for the full equation of the line through A and B, the slope calculator.
Frequently asked questions
What is the midpoint formula?
M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). Add the two x-coordinates and halve the sum, then do the same for y, and for z in 3D. The midpoint of (−2, 3) and (6, 7) is (2, 5).
How do I find an endpoint when I know the midpoint?
Double the midpoint and subtract the known endpoint, one coordinate at a time: B = 2M − A. If A is (7, 2) and M is (3, −1), then B = (6 − 7, −2 − 2) = (−1, −4).
What is a perpendicular bisector?
It is the line that crosses a segment at its midpoint at a right angle. Every point on it is the same distance from both endpoints, which makes it useful for locating a circle's center or a spot equally far from two places. The calculator gives its equation for 2D segments.
Is the midpoint just an average?
Yes. Each coordinate of the midpoint is the mean of the endpoints' coordinates, which is why the formula works in any number of dimensions. Dividing a segment in another ratio, such as one third of the way along, uses a weighted average instead.
Why is there no perpendicular bisector in 3D mode?
In space, infinitely many lines meet a segment at right angles at its midpoint. Together they form a plane, the perpendicular bisector plane, which a single line equation can't describe. The calculator therefore shows the bisector only for 2D segments.