The Midpoint Formula
The midpoint formula M = ((x₁+x₂)/2, (y₁+y₂)/2): what it means, a short proof with similar triangles, the 3D and number-line versions, and worked examples.
The Midpoint Formula: Use It in 30 Seconds
You have two points on a map: (2, 3) and (6, 7). The exact center between them is (4, 5). That is the midpoint, and the midpoint formula is how you find it every time. Add the x-coordinates, divide by two. Add the y-coordinates, divide by two. That is the entire formula for 2D. For 3D, you add the z-coordinate, same operation, same logic.
You will see why it works, how to use it when coordinates are negative or fractional, and where it stops applying (the moment someone hands you three points). No history, no filler, no high-dimensional theory you do not need.
What Is a Midpoint? Definition Geometry Students Need
A midpoint is the exact center point of a line segment. That is the only definition you need. It lies halfway between the two endpoints, dividing the segment into two equal lengths. In coordinate geometry, the midpoint is the point whose coordinates are the arithmetic mean of the two endpoints.
The midpoint definition geometry standard (Common Core HSG-GPE.B.6) says the same thing: "Find the point on a directed line segment between two given points that partitions the segment in a given ratio." The midpoint is the 1:1 ratio case. If you are studying for a test, this is the definition you commit to memory because it is the one the exam expects.
What a midpoint is not: it is not the center of a triangle, not the average of three or more points, not a line or a ray. It is a single point on a line segment between exactly two endpoints. The most common mistake students make is calling the average of three points a midpoint. That is a centroid, and they are different concepts.
The Formula: 1D, 2D, and 3D
The pattern is the same in every dimension: average the coordinates of the two endpoints.
1D Number Line
On a number line, the midpoint between 2 and 10 is (2 + 10) / 2 = 6. One dimension, one division. That is the whole concept.
2D Coordinate Plane
For points A(x₁, y₁) and B(x₂, y₂), the midpoint M is:
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2)
Example: A(2, 3) and B(6, 7). Add x: 2 + 6 = 8, divide by 2 → 4. Add y: 3 + 7 = 10, divide by 2 → 5. Midpoint: (4, 5).
3D Space
For points A(x₁, y₁, z₁) and B(x₂, y₂, z₂), add the z-coordinate:
M = ((x₁ + x₂) / 2, (y₁ + y₂) / 2, (z₁ + z₂) / 2)
Example: A(1, 2, 3) and B(7, 8, 9). x: (1+7)/2 = 4. y: (2+8)/2 = 5. z: (3+9)/2 = 6. Midpoint: (4, 5, 6).
| Dimension | Formula | Example | When to Use |
|---|---|---|---|
| 1D | M = (x₁ + x₂) / 2 | (2 + 10) / 2 = 6 | Number line, time intervals |
| 2D | M = ((x₁+x₂)/2, (y₁+y₂)/2) | (2+6)/2 = 4, (3+7)/2 = 5 → (4,5) | Coordinate plane, maps |
| 3D | M = ((x₁+x₂)/2, (y₁+y₂)/2, (z₁+z₂)/2) | (1+7)/2 = 4, (2+8)/2 = 5, (3+9)/2 = 6 → (4,5,6) | 3D geometry, physics |
Why It Works: Averaging and a Proof
The formula works because coordinates are linear. On a number line, the point halfway between two numbers is their average. That is the definition of an arithmetic mean. In 2D, each coordinate axis is independent, so the x-coordinate of the midpoint is the average of the x-coordinates, and the y-coordinate is the average of the y-coordinates.
The proof is straightforward. Let A(x₁, y₁) and B(x₂, y₂) be the endpoints of a line segment. The midpoint M must satisfy that the distance from A to M equals the distance from M to B. Because coordinates are linear, this condition reduces to two equations: x-coordinate from A to M equals x-coordinate from M to B, and same for y. Solving gives x = (x₁ + x₂)/2 and y = (y₁ + y₂)/2.
Proof Diagram
Draw a line segment on a coordinate plane. Mark the endpoints. Draw a vertical line from each endpoint to the x-axis. The midpoint's x-coordinate is the average of those two intercepts. Same for the y-axis. The diagram makes the averaging visual: the midpoint sits at the crossing of the two average lines.
This proof works for any number of dimensions because each coordinate axis is independent. The distance formula verifies the result: measure from A to M and from M to B; they are equal.
Worked Examples: Three You Will See on Tests
Example 1: Basic 2D with Positive Coordinates
Find the midpoint between (3, 1) and (9, 5).
x: (3 + 9) / 2 = 12 / 2 = 6
y: (1 + 5) / 2 = 6 / 2 = 3
Midpoint: (6, 3).
Example 2: 2D with Negative Coordinates
Find the midpoint between (-4, 7) and (2, -5).
x: (-4 + 2) / 2 = -2 / 2 = -1
y: (7 + (-5)) / 2 = 2 / 2 = 1
Midpoint: (-1, 1). The sign does not break the formula; treat it as normal addition.
Example 3: 3D with Fractional Coordinates
Find the midpoint between (0.5, 1.2, -3) and (2.5, 3.8, 5).
x: (0.5 + 2.5) / 2 = 3.0 / 2 = 1.5
y: (1.2 + 3.8) / 2 = 5.0 / 2 = 2.5
z: (-3 + 5) / 2 = 2 / 2 = 1
Midpoint: (1.5, 2.5, 1).
Common failure on tests: forgetting to divide by 2. Students add the coordinates and write the sum as the midpoint. Always divide by 2, and if the result is not a whole number, leave it as a decimal or fraction.
Midpoint vs Centroid: The One Distinction You Must Not Confuse
The midpoint is defined for exactly two points. The centroid is the average of three or more points (or of all points in a shape). They are not interchangeable.
If you have three points that form a triangle, the centroid is the average of the three vertices. The midpoint is a point on one side of the triangle. If you have three collinear points, say (1,1), (3,3), and (5,5), they do not form a triangle, so there is no centroid. The midpoint of the segment from (1,1) to (5,5) is (3,3), which happens to be one of the original points, but calling it the centroid of a triangle would be wrong because there is no triangle.
It said the average of three or more points is a midpoint. That is the single most persistent error in teaching this topic. The centroid formula for a triangle uses a different calculation (area-weighted), and the section formula handles ratios other than 1:1. If you need to find the center of a polygon, you want the centroid, not the midpoint.
More in this guide
How to Find the Equation of a Perpendicular BisectorThe Midpoint Formula Economics Guide.
Common Questions
When do I use the 2D formula vs the 3D formula?
Use 2D when the problem gives two coordinates with x and y. Use 3D when it gives x, y, and z. If the problem is on a coordinate plane, it is 2D. If it is in space (a box, a 3D graph), it is 3D.
What happens if the two points are the same?
If both points are identical, the midpoint is that same point. For example, (3,5) and (3,5) give (3,5). The segment has zero length, but the formula still works.
Can I use the midpoint formula for three or more points?
No. Averaging three or more points gives the centroid, not the midpoint. If a problem asks for the midpoint of three points, it is either misusing the term or expecting you to find the centroid. The midpoint only exists for two points.
How do I find a missing endpoint when I have the midpoint and one endpoint?
Use the missing endpoint formula: x₂ = 2Mx - x₁, y₂ = 2My - y₁. For example, if midpoint is (5,4) and one endpoint is (3,2), then x₂ = 2(5) - 3 = 7, y₂ = 2(4) - 2 = 6. The missing endpoint is (7,6).
Does the midpoint formula work with decimals and fractions?
Yes. Treat decimals and fractions as normal numbers. For fractions, add them and divide by 2, or multiply by 1/2. For decimals, the arithmetic is the same.
What is the difference between a midpoint and a bisector?
A midpoint is a point. A bisector is a line, ray, or segment that passes through the midpoint and divides something into two equal parts. The perpendicular bisector is a line through the midpoint at a right angle to the segment.
Does the midpoint formula apply to geographic coordinates (latitude and longitude)?
Not directly. Averaging latitude and longitude gives a point, but on a sphere (Earth), the true halfway point along the surface is found using the Haversine formula. The simple average works only for very short distances where the Earth's curvature is negligible.