Math Core

The midpoint is the point that splits a segment into two equal halves. It shows up everywhere in geometry: the center of a circle is the midpoint of every diameter, the diagonals of a parallelogram meet at their midpoints, and a midpoint is often the first thing you construct in a proof.

What a midpoint is

Definition

Midpoint

The midpoint of $\overline{AB}$ is the point $M$ on $\overline{AB}$ with $AM = MB$ . It divides the segment into two congruent segments, $\overline{AM} \cong \overline{MB}$ .

Because the two halves are equal, each one is half of the whole:

$AM = MB = \tfrac12 AB \qquad \text{and} \qquad AB = 2 \cdot AM.$

A line, ray or segment that passes through the midpoint of a segment is called a segment bisector. A segment has exactly one midpoint, but it has infinitely many bisectors, since any line through $M$ other than line $AB$ itself will do.

Line k bisects segment AB at its midpoint M. The matching tick marks show AM = MB.

Midpoints on a number line

If $A$ and $B$ have coordinates $a$ and $b$ , the midpoint is halfway between them, which is their average:

$M = \frac{a + b}{2}.$

For example, the midpoint of $-4$ and $10$ is $\dfrac{-4 + 10}{2} = \dfrac{6}{2} = 3$ . Check: $3$ is $7$ units from each endpoint.

The Midpoint Formula

In the coordinate plane, average the $x$ -coordinates and average the $y$ -coordinates separately. The midpoint is halfway across and halfway up.

The Midpoint Formula

The midpoint of the segment joining $(x_1, y_1)$ and $(x_2, y_2)$ is

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

M is halfway between A and B in both directions: 4 units across and 2 units up from A, then 4 across and 2 up again to B.Open in grapher →

Common mistake

Two common slips: subtracting instead of adding the coordinates (that gives half the change, not the midpoint), and mixing an $x$ with a $y$ . Add the two $x$ 's and halve, then add the two $y$ 's and halve. The distance formula subtracts; the midpoint formula adds.

Worked example: Finding a midpoint

Find the midpoint of the segment joining $(-3, -7)$ and $(4, 2)$ .

$M = \left( \frac{-3 + 4}{2}, \ \frac{-7 + 2}{2} \right) = \left( \frac12, \ -\frac52 \right).$

So $M = (0.5, -2.5)$ . A midpoint doesn't have to have whole-number coordinates.

Working backward: finding an endpoint

Sometimes you know one endpoint and the midpoint, and you need the other endpoint. Think of it as a trip: going from $A$ to $M$ is half the journey, so repeat the same step once more to reach $B$ .

Worked example: Finding the other endpoint

$M(1, 4)$ is the midpoint of $\overline{AB}$ , and $A = (-3, 7)$ . Find $B$ .

Method 1 (repeat the step). From $A$ to $M$ , $x$ goes from $-3$ to $1$ (up $4$ ) and $y$ goes from $7$ to $4$ (down $3$ ). Take the same step from $M$ : $x = 1 + 4 = 5$ and $y = 4 - 3 = 1$ . So $B = (5, 1)$ .

Method 2 (use the formula). Let $B = (x, y)$ . Then

$\frac{-3 + x}{2} = 1 \implies x = 5, \qquad \frac{7 + y}{2} = 4 \implies y = 1.$

Check: the midpoint of $(-3, 7)$ and $(5, 1)$ is $\left(\dfrac{2}{2}, \dfrac{8}{2}\right) = (1, 4)$ .

Tip

A quick shortcut for the missing endpoint: $B = 2M - A$ , done coordinate by coordinate. Here $B = (2 \cdot 1 - (-3), \ 2 \cdot 4 - 7) = (5, 1)$ . To check any midpoint answer, use the distance formula: $M$ should be exactly the same distance from both endpoints. Here $AM = \sqrt{4^2 + 3^2} = 5$ and $MB = \sqrt{4^2 + 3^2} = 5$ .

Midpoints and algebra

Since a midpoint makes two equal lengths, " $M$ is the midpoint" gives you an equation to solve.

Worked example: A midpoint equation

$M$ is the midpoint of $\overline{AB}$ , with $AM = 3x + 1$ and $MB = 5x - 7$ . Find $AB$ .

The halves are equal: $3x + 1 = 5x - 7$ , so $8 = 2x$ and $x = 4$ . Then $AM = 3(4) + 1 = 13$ (and $MB = 5(4) - 7 = 13$ as a check). The whole segment is twice a half: $AB = 2 \cdot 13 = 26$ .

Read carefully whether a problem gives you a half ( $AM$ ) or the whole ( $AB$ ). If $AM = x + 4$ and $AB = 3x - 2$ , the equation is $2(x + 4) = 3x - 2$ , not $x + 4 = 3x - 2$ .

Practice

Practice 1

On a number line, $A$ is at $-9$ and $B$ is at $3$ . What is the coordinate of the midpoint of $\overline{AB}$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 2

Find the midpoint of the segment joining $(4, -2)$ and $(10, 6)$ .

Enter a point like (2, -3)

Practice 3

Find the midpoint of the segment joining $(-5, 3)$ and $(2, -4)$ .

Enter a point like (2, -3)

Practice 4

Line $k$ passes through the midpoint $M$ of $\overline{PQ}$ . Which statement must be true?

Practice 5

$M(2, -1)$ is the midpoint of $\overline{AB}$ , and $A = (-4, 3)$ . Find the coordinates of $B$ .

Enter a point like (2, -3)

Practice 6

$M$ is the midpoint of $\overline{AB}$ , with $AM = 4x - 3$ and $MB = 2x + 9$ . Find $AB$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

$M$ is the midpoint of $\overline{AB}$ , with $AM = x + 5$ and $AB = 5x - 8$ . Find $x$ .

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 8

The midpoint of the segment joining $(a, 3)$ and $(7, b)$ is $(4, -1)$ . Find $a$ and $b$ , and give your answer as the ordered pair $(a, b)$ .

Enter a point like (2, -3)