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# Midpoint Formula — Definition, Derivation, and Examples

## TL;DR

The midpoint formula finds the exact centre of a line segment by averaging the coordinates of its two endpoints: M=(x1+x2/2,y1+y2/2). This article defines the midpoint, derives the formula from a simple average, and works through examples with positive, negative, and fractional coordinates.

The **midpoint formula** gives the coordinates of the point exactly halfway between two endpoints. For points A(x1, y1) and B(x2, y2), the midpoint is the average of the two x-coordinates paired with the average of the two y-coordinates:

$$M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$$

Here (x1, y1) is one endpoint, (x2, y2) is the other, and M is the midpoint. The formula divides the segment in a 1:1 ratio — two equal halves. It lives on the [coordinate plane](/content/math/geometry/coordinate-plane/index.html), the same grid where you plot points and lines.

## Where Does the Midpoint Formula Come From?

The formula is not a rule to memorise blindly — it is just an average, applied to each coordinate separately.

Start on a number line. The midpoint of 5 and 9 is their average: \( \frac{5 + 9}{2} = 7 \), and 7 sits exactly between them. The coordinate plane is two number lines at right angles, so you average each direction on its own — the x-values give the horizontal centre, the y-values give the vertical centre.

The derivation is simple: the midpoint is the average point. The closely related [distance formula](/content/math/formula/distance-formula/index.html) measures how far apart the endpoints are, while the midpoint formula finds the centre between them, and the two are often used together.

## The Midpoint On A Number Line

Before the two-coordinate version, it helps to see the one-dimensional case. On a number line, the midpoint of two values a and b is simply their average:

\( \text{midpoint} = \frac{a + b}{2} \)

So the midpoint of 3 and 11 is \( \frac{3 + 11}{2} = 7 \). The coordinate-plane formula is this same average applied twice — once horizontally, once vertically.

## Formulas Related To The Midpoint

The midpoint is the simplest member of a small family of "in-between point" formulas.

- **Section formula.** The midpoint divides a segment in the ratio 1:1. The more general section formula finds a point dividing a segment in any ratio m:n and reduces to the midpoint formula when m=n.
- **Centroid of a triangle.** Averaging three vertices instead of two gives the centroid, \( \left(\frac{x_1 + x_2 + x_3}{3}, \frac{y_1 + y_2 + y_3}{3}\right) \) — the same averaging idea extended to a triangle.

## Examples Of The Midpoint Formula

### Example 1

**Find the midpoint of A(2,1) and B(6,3).**

Average the x-values and the y-values.

M=(2+6/2, 1+3/2)=(8/2, 4/2)=(4, 2)

**Final answer:** The midpoint is (4, 2).

### Example 2

**A student finds the midpoint of (−3,0) and (0,−5) by subtracting the coordinates instead of averaging. Where does this go wrong?**

The intuitive move, when one coordinate is negative, is to "find the gap" by subtracting — giving something like (−3,5). But that does not land between them; it measures a difference, not a centre. The correct method is to average the coordinates.

M=(−3+0/2,0+(−5)/2)=(−3/2,−5/2)=(-1.5,−2.5)

**Final answer:** The midpoint is (−1.5,−2.5).

### Example 3

**The endpoints of a circle's diameter are (2,−3) and (−6,5). Find the centre of the circle.**

M=(2+(−6)/2,−3+5/2)=(−4/2,2/2)=(-2, 1)

**Final answer:** The centre is (−2,1).

### Example 4

**Find the midpoint of A(6,8) and B(3,1).**

M=(6+3/2,8+1/2)=(9/2,9/2)=(4.5, 4.5)

**Final answer:** The midpoint is (4.5,4.5).

### Example 5

**The midpoint of segment PQ is (4,4). One endpoint is P(1,2). Find the other endpoint Q.**

Set the average of P and Q equal to the midpoint:

1+x/2=4 ⟹ 1+x=8 ⟹ x=7

2+y/2=4 ⟹ 2+y=8 ⟹ y=6

**Final answer:** Q=(7,6).

### Example 6

**A triangle has vertices at A(0,0), B(8,0), and C(4,6). Find the midpoint of side AB, then check it lies on the line y=0.**

Midpoint of AB:

M=(0+8/2,0+0/2)=(4,0)

Since its y-coordinate is 0, the midpoint lies on the x-axis, y=0.

**Final answer:** The midpoint of AB is (4,0), which lies on y=0.

## Why The Halfway Point Earns Its Keep

The midpoint formula looks small, but it is a building block. Surveyors, mapmakers, and computer-graphics engineers all rely on it to find the exact centre of a span.

- **Finding a centre.** The centre of a circle is the midpoint of its diameter; the centre of a rectangle is the midpoint of its diagonal.
- **Bisecting a segment.** The midpoint is where a [perpendicular bisector](/content/math/geometry/perpendicular-bisectors/index.html) crosses a segment.
- **Smooth motion in graphics.** Animation and games interpolate positions using midpoints.

## Mistakes With The Midpoint Formula

### Mistake 1: Subtracting coordinates instead of averaging

**Don't do this:** Compute x2−x1 and call it the midpoint.

**The correct way:** Always add the two coordinates and divide by two.

### Mistake 2: Mixing up which coordinate goes with which

**Don't do this:** Average x1 with y2. The x-coordinates average together, and the y-coordinates average together.

### Mistake 3: Forgetting the formula reverses for "find the other endpoint" problems

**Don't do this:** Plug the midpoint straight into the formula.

**The correct way:** Set the average of the known endpoint and the unknown equal to the given midpoint, then solve.

## Conclusion

- The **midpoint formula** is M=(x1 + x2/2, y1 + y2/2), averaging the x-values and the y-values.
- It comes straight from the idea of an average, applied to each coordinate separately.
- A fractional or negative midpoint is normal; it simply means the midpoint falls between gridlines or in another quadrant.

## Practice These To Solidify Your Understanding

1. Find the midpoint of (1,7) and (5,3). _(Answer: (3,5).)_
2. Find the midpoint of (−4,2) and (2,−6). _(Answer: (−1,−2).)_
3. The midpoint of RS is (0,0) and R=(−3,4). Find S. _(Answer: (3,−4).)_