The midpoint of a line segment is the point exactly halfway between its two endpoints. To find it, add the two x-coordinates and divide by 2, then add the two y-coordinates and divide by 2. In other words, the midpoint is the mean of the two endpoints' coordinates: M = ((x₁ + x₂)/2, (y₁ + y₂)/2).
Why does averaging the coordinates give the midpoint?
The x-coordinate of the midpoint is exactly halfway between x₁ and x₂ on the horizontal axis; this is the mean of x₁ and x₂. The same logic applies vertically. You can think of it as splitting the journey from one point to the other into two equal halves — first in the x-direction, then in the y-direction.
How do you find the midpoint of a line segment?
Formula:
M = ((x₁ + x₂)/2, (y₁ + y₂)/2)
Method — three steps:
- Add the two x-coordinates.
- Divide by 2.
- Repeat for the y-coordinates.
Worked example 1:
Find the midpoint of the segment joining A(2, 6) and B(8, 4).
x-coordinate of M: (2 + 8)/2 = 10/2 = 5
y-coordinate of M: (6 + 4)/2 = 10/2 = 5
M = (5, 5)
Plot check: A is at (2, 6), B is at (8, 4), M is at (5, 5). The distance AM should equal MB. AM: horizontal = 3, vertical = 1; MB: horizontal = 3, vertical = 1. ✓
Worked example 2:
Find the midpoint of C(−3, 7) and D(5, −1).
x: (−3 + 5)/2 = 2/2 = 1
y: (7 + (−1))/2 = 6/2 = 3
M = (1, 3)
Midpoints with negative coordinates
Negative coordinates follow the same formula — just take care with the arithmetic.
| Segment | x calculation | y calculation | Midpoint |
|---|---|---|---|
| (4, 8) to (10, 2) | (4+10)/2 = 7 | (8+2)/2 = 5 | (7, 5) |
| (−6, 3) to (2, 7) | (−6+2)/2 = −2 | (3+7)/2 = 5 | (−2, 5) |
| (−4, −2) to (6, −8) | (−4+6)/2 = 1 | (−2+(−8))/2 = −5 | (1, −5) |
| (0, 5) to (−7, 3) | (0+(−7))/2 = −3.5 | (5+3)/2 = 4 | (−3.5, 4) |
Notice that midpoints can have non-integer coordinates even when the endpoints are both integers — this is correct. For example: segment from (1, 0) to (2, 5) gives midpoint (1.5, 2.5).
Worked example 3 — from a graph
A line is drawn on a grid from P(1, 5) to Q(7, 3). Mark the midpoint.
x: (1 + 7)/2 = 4
y: (5 + 3)/2 = 4
Midpoint = (4, 4)
Count along the segment: from P move 3 right and 1 down — you land at (4, 4); then another 3 right and 1 down lands at Q(7, 3). ✓ The midpoint is equidistant from both ends.
How do you find a missing endpoint given the midpoint?
If you know one endpoint and the midpoint, you can find the other endpoint by reversing the formula.
Formula for the unknown endpoint:
If M = ((x₁ + x₂)/2, (y₁ + y₂)/2), then:
x₂ = 2 × x_M − x₁ and y₂ = 2 × y_M − y₁
Worked example 4:
The midpoint of segment AB is M(3, −1). One endpoint is A(−1, 5). Find B.
x_B = 2 × 3 − (−1) = 6 + 1 = 7
y_B = 2 × (−1) − 5 = −2 − 5 = −7
B = (7, −7)
Check: midpoint of A(−1, 5) and B(7, −7):
x: (−1 + 7)/2 = 3 ✓; y: (5 + (−7))/2 = −1 ✓.
Midpoints in geometry problems
Midpoints appear in KS3 geometry in several contexts: finding the centre of a line segment, constructing perpendicular bisectors (the perpendicular bisector passes through the midpoint), and identifying special points in triangles (such as the centroid, which is the average of all three vertex coordinates — an extension of the midpoint idea).
Frequently asked questions
Does it matter which point I call (x₁, y₁) and which I call (x₂, y₂)?
No. Addition is commutative, so (x₁ + x₂)/2 = (x₂ + x₁)/2. Label either point as the first and either as the second — the midpoint is the same.
What if the segment is on just one axis?
The formula still works. For a horizontal segment from (2, 4) to (8, 4), the y-coordinates are the same. Midpoint: ((2+8)/2, (4+4)/2) = (5, 4). Notice the midpoint lies on the same horizontal line, as expected.
Can I find the midpoint without the formula — just by looking at the grid?
For simple integer coordinates, yes — count the total horizontal distance, halve it, and count the total vertical distance, halve it. This is effective on a clearly drawn grid, but the formula is faster and more reliable for coordinates with large or negative values, so it is worth practising both approaches.
How does the midpoint link to finding the equation of a perpendicular bisector?
A perpendicular bisector of a segment is a line that (1) passes through the midpoint and (2) makes a right angle with the original segment. To find its equation: find the midpoint (using this method), find the gradient of the original segment, then use the perpendicular gradient rule (negative reciprocal) and the midpoint as a known point. This is a GCSE-level extension of the KS3 midpoint idea.
Work through coordinate geometry with Professor Pi at aitutors.me.