# What is the midpoint?

Finds the midpoint of two points on a plane and the distance between them, or the other endpoint from one endpoint and the midpoint.

- Page: https://www.acalculator.org/math/midpoint-calculator
- JSON spec: https://www.acalculator.org/math/midpoint-calculator.json
- Version: 78e0497537d5

## Default answer

Example with the default inputs (What do you want to find? Midpoint, Point A: x 2, Point A: y 3, Point B: x 8, Point B: y 5): The midpoint of (2, 3) and (8, 5) is (5, 4).

## Inputs

| Key | Label | Description |
| --- | --- | --- |
| find | What do you want to find? | The midpoint of two points, or the second endpoint from the first endpoint and the midpoint. |
| x1 | Point A: x | The x-coordinate of the first point, A. |
| y1 | Point A: y | The y-coordinate of the first point, A. |
| x2 | Point B: x | The x-coordinate of the second point, B. |
| y2 | Point B: y | The y-coordinate of the second point, B. |
| mx | Midpoint M: x | The x-coordinate of the midpoint, M. |
| my | Midpoint M: y | The y-coordinate of the midpoint, M. |

## Outputs

| Key | Label | Description |
| --- | --- | --- |
| mx | Midpoint: x | The x-coordinate of the midpoint: (x₁ + x₂) ÷ 2. |
| my | Midpoint: y | The y-coordinate of the midpoint: (y₁ + y₂) ÷ 2. |
| x2 | Point B: x | The x-coordinate of the other endpoint: 2 × Mx − x₁. |
| y2 | Point B: y | The y-coordinate of the other endpoint: 2 × My − y₁. |
| distance | Distance from A to B | The straight-line length of the segment AB: √((x₂ − x₁)² + (y₂ − y₁)²). |

## Method

M = ((x₁ + x₂) ÷ 2, (y₁ + y₂) ÷ 2); from A and M, B = (2 × Mx − x₁, 2 × My − y₁).

## Assumptions

- Points on a flat plane with x and y coordinates in the same unit.
- The distance is the straight-line (Euclidean) distance from A to B.

## Worked examples

1. find = midpoint, x1 = 2, y1 = 3, x2 = 8, y2 = 5 gives mx = 5, my = 4, distance = 6.324555. Source: OpenStax, College Algebra 2e, §2.1 The Rectangular Coordinate Systems and Graphs (distance and midpoint formulas). https://openstax.org/books/college-algebra-2e/pages/2-1-the-rectangular-coordinate-systems-and-graphs.
2. find = midpoint, x1 = -4, y1 = 3, x2 = 6, y2 = -7 gives mx = 1, my = -2, distance = 14.142136. Source: OpenStax, College Algebra 2e, §2.1 The Rectangular Coordinate Systems and Graphs (distance and midpoint formulas). https://openstax.org/books/college-algebra-2e/pages/2-1-the-rectangular-coordinate-systems-and-graphs.
3. find = midpoint, x1 = 0.1, y1 = 0.2, x2 = 0.3, y2 = 0.4 gives mx = 0.2, my = 0.3. Source: OpenStax, College Algebra 2e, §2.1 The Rectangular Coordinate Systems and Graphs (distance and midpoint formulas). https://openstax.org/books/college-algebra-2e/pages/2-1-the-rectangular-coordinate-systems-and-graphs.
4. find = endpoint, x1 = 2, y1 = 3, mx = 5, my = 4 gives x2 = 8, y2 = 5. Source: OpenStax, College Algebra 2e, §2.1 The Rectangular Coordinate Systems and Graphs (distance and midpoint formulas). https://openstax.org/books/college-algebra-2e/pages/2-1-the-rectangular-coordinate-systems-and-graphs.

## FAQ

### What is a midpoint in geometry?

A midpoint is the exact center point between two given points on a line, a plane, or in space. It's the point that divides the line segment connecting two points into two equal parts. In 2D coordinates, the midpoint is found by averaging the x-coordinates and y-coordinates of the two points.

### How do you calculate the midpoint between two points?

The midpoint formula for two points (x₁, y₁) and (x₂, y₂) is:

M = ((x₁ + x₂)/2, (y₁ + y₂)/2)

This means you:

1. Add the x-coordinates and divide by 2
2. Add the y-coordinates and divide by 2
3. The result gives you the midpoint coordinates

### Can I use this calculator for 3D coordinates?

This calculator is specifically designed for 2D coordinates (x, y). For 3D coordinates, you would need to extend the formula to include the z-coordinate: M = ((x₁ + x₂)/2, (y₁ + y₂)/2, (z₁ + z₂)/2).

### What is the distance between two points?

The distance between two points (x₁, y₁) and (x₂, y₂) is calculated using the distance formula:

d = √[(x₂ - x₁)² + (y₂ - y₁)²]

This is based on the Pythagorean theorem and gives you the straight-line distance between the two points.

### Why is the midpoint important in geometry?

The midpoint is crucial in geometry because it helps in dividing line segments equally, finding centers of shapes, constructing perpendicular bisectors, and solving various geometric problems. It's also used in computer graphics, navigation systems, and many real-world applications where you need to find the center point between two locations.

### Can the midpoint be outside the line segment?

No, the midpoint always lies exactly on the line segment connecting the two points. It's impossible for the midpoint to be outside the line segment because it's defined as the point that divides the segment into two equal parts.

### How accurate are the calculations?

The calculator works in 64-bit floating point and shows up to 6 decimal places for the coordinates and the distance. The calculations use standard mathematical formulas and are suitable for most practical applications including academic work, engineering, and design.

### What are some real-world applications of midpoint calculations?

Midpoint calculations are used in many real-world scenarios:

- **Navigation:** Finding the center point between two locations
- **Computer Graphics:** Creating smooth curves and animations
- **Architecture:** Determining center points for design elements
- **Surveying:** Finding midpoints for property boundaries
- **Game Development:** Calculating center points for game objects
- **Data Analysis:** Finding central tendencies in spatial data

### How do I find an endpoint from the midpoint?

If you know one endpoint A (x₁, y₁) and the midpoint M, the other endpoint B is (2 × Mx − x₁, 2 × My − y₁). For example, with A = (2, 3) and M = (5, 4), B = (2 × 5 − 2, 2 × 4 − 3) = (8, 5). Pick Endpoint under "What do you want to find?" to do this.

## Sources

- OpenStax, College Algebra 2e, §2.1 The Rectangular Coordinate Systems and Graphs (distance and midpoint formulas). https://openstax.org/books/college-algebra-2e/pages/2-1-the-rectangular-coordinate-systems-and-graphs