OneTaskBy Tim King · Updated
Midpoint Calculator
Missing Endpoint From a Midpoint, Working Shown for Any Line Segment
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How the Midpoint Calculator works
This midpoint calculator finds the point exactly halfway along a line segment from its two endpoints. Switch the mode and it works backwards instead, giving the missing endpoint when you know one end and the midpoint. Each step is shown, along with the segment length and the points at thirds.
The midpoint sits halfway between the two ends in both directions at once. Halfway across in x is the average of the two x values, and halfway up in y is the average of the two y values. That is all there is to it, and it works the same way for negative numbers and decimals.
How the midpoint calculator works
The formula is M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). To find a missing endpoint B from A and M, rearrange it: B = (2Mx − Ax, 2My − Ay). In words, double the midpoint and subtract the end you know.
First example. The midpoint of (2, 3) and (8, 7) is ((2 + 8) ÷ 2, (3 + 7) ÷ 2) = (5, 5).
Second example. One end is (1, 2) and the midpoint is (4, 6). The other end is (2 × 4 − 1, 2 × 6 − 2) = (7, 10).
How many points split a segment into equal parts? Two points cut it into thirds. From (0, 0) to (9, 6), they sit at (3, 2) and (6, 4), which the calculator also lists.
Midpoint and length of 24 line segments
| Point A | Point B | Midpoint | Length |
|---|---|---|---|
| (0, 0) | (4, 6) | (2, 3) | 7.211 |
| (2, 3) | (8, 7) | (5, 5) | 7.211 |
| (-4, 1) | (6, 5) | (1, 3) | 10.770 |
| (1, 1) | (2, 2) | (1.5, 1.5) | 1.414 |
| (-3, -5) | (3, 5) | (0, 0) | 11.662 |
| (0, 10) | (10, 0) | (5, 5) | 14.142 |
| (5, 2) | (9, 2) | (7, 2) | 4.000 |
| (-2, 7) | (-2, -1) | (-2, 3) | 8.000 |
| (1.5, 2) | (4.5, 6) | (3, 4) | 5.000 |
| (-6, -4) | (-2, 8) | (-4, 2) | 12.649 |
| (3, 9) | (11, 1) | (7, 5) | 11.314 |
| (0, 0) | (7, 3) | (3.5, 1.5) | 7.616 |
| (12, 5) | (-4, 9) | (4, 7) | 16.492 |
| (-1, -1) | (-9, -7) | (-5, -4) | 10.000 |
| (6, 6) | (6, 6) | (6, 6) | 0.000 |
| (2, -8) | (10, 4) | (6, -2) | 14.422 |
| (-5, 0) | (0, 5) | (-2.5, 2.5) | 7.071 |
| (100, 50) | (20, 30) | (60, 40) | 82.462 |
| (7, 3) | (1, 11) | (4, 7) | 10.000 |
| (-10, 4) | (10, -4) | (0, 0) | 21.541 |
| (0.5, 0.5) | (3.5, 2.5) | (2, 1.5) | 3.606 |
| (4, 4) | (13, 16) | (8.5, 10) | 15.000 |
| (-7, 2) | (3, -6) | (-2, -2) | 12.806 |
| (9, 0) | (0, 12) | (4.5, 6) | 15.000 |
Tips
- Check with the distance. The midpoint should be the same distance from each end.
- Halves are normal. Odd sums give .5 coordinates; that is correct, not a rounding error.
- The perpendicular bisector passes through the midpoint at the negative reciprocal slope.
FAQ
What is the midpoint formula?
Average the x coordinates and average the y coordinates: M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2). For the segment from (−4, 1) to (6, 5), that is (2 ÷ 2, 6 ÷ 2) = (1, 3). It is the same as finding the mean of two numbers, done once for each axis.
How do you find an endpoint from the midpoint?
Double each midpoint coordinate and subtract the known endpoint. If A is (3, −2) and M is (5, 1), then B is (2 × 5 − 3, 2 × 1 + 2) = (7, 4). Check it by finding the midpoint of A and B, which should give M back.
Is the midpoint the same as the centre of a circle?
When the segment is a diameter, yes. The midpoint of any diameter is the centre, so if you know two opposite points on a circle you can find its centre with this calculator, and half the segment length gives the radius.
Does the midpoint formula work in 3D?
Yes. Average the z values too: M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2, (z₁ + z₂) ÷ 2). The midpoint of (0, 2, 4) and (6, 8, 10) is (3, 5, 7). Each axis is handled separately, so adding more dimensions just adds more averages.
See also: all Algebra and Geometry tools, Arithmetic, Number Theory, Statistics, Trigonometry and Calculus.
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