What is a midpoint?
The midpoint of a line segment is the point exactly halfway between its two endpoints. It splits the segment into two equal parts. On a coordinate plane, its x-coordinate is the average of the two x-coordinates, and its y-coordinate is the average of the two y-coordinates.
Midpoints show up in geometry class, in graphing, in design layouts and anywhere you need the centre of a line.
The midpoint formula
For points (x1, y1) and (x2, y2):
M = ((x1 + x2) / 2, (y1 + y2) / 2)
In 3D, just add a third coordinate: the z-value of the midpoint is (z1 + z2) / 2. In other words, the midpoint is found by averaging each coordinate separately.
How to find a midpoint step by step
- Write down both points.
- Add the two x-values and divide by 2.
- Add the two y-values and divide by 2.
- Write the results as an ordered pair.
How to find a missing endpoint
Sometimes you know one endpoint and the midpoint, and you need the other endpoint. Because the midpoint is the average, rearrange the formula: the missing coordinate is 2 ร midpoint โ known coordinate.
A quick check is to put the answer back into the midpoint formula and see that you get the midpoint you started with.
Distance between the two points
This calculator also gives the distance between the endpoints using the distance formula, which comes from the Pythagorean theorem. The distance from either endpoint to the midpoint is exactly half of that total, which is a handy way to check your work.
Common mistakes
- Forgetting to divide by 2, which gives the sum instead of the average.
- Dropping a negative sign. Put negative numbers in brackets when you work by hand.
- Mixing up the order of x and y values.
- Using the midpoint formula to find a point one third of the way along. That needs a different ratio.
For right-triangle problems, the Pythagorean theorem calculator is a good companion.
Midpoints in 3D
The idea is identical in three dimensions. For A = (1, 2, 3) and B = (5, 8, 9), the midpoint is ((1 + 5)/2, (2 + 8)/2, (3 + 9)/2) = (3, 5, 6). The distance is โ(4ยฒ + 6ยฒ + 6ยฒ) = โ88 โ 9.38. Choose a 3D mode in the calculator and a z-coordinate box appears.
Where midpoints are used
- Geometry: finding the centre of a segment, the centre of a circle from the ends of a diameter, or the median of a triangle.
- Design and layout: placing an object exactly between two others.
- Maps and games: finding a meeting point between two locations.
- Data: the midpoint of a range is the average of its low and high values.
Using a midpoint to find the centre of a circle
If you know the two ends of a circle's diameter, their midpoint is the centre. For ends (โ4, 1) and (6, 5), the centre is ((โ4 + 6)/2, (1 + 5)/2) = (1, 3). The radius is half the distance between them. The missing endpoint mode helps in the reverse case, where you know the centre and one end of the diameter and need the other end.