The midpoint of a line segment

What the midpoint is

The midpoint is the point exactly halfway along the segment joining two points — the same distance from each end.
Its co-ordinates are the average of the two end co-ordinates: average the $x$ -values, then average the $y$ -values.

The midpoint formula

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

$\frac{x_1 + x_2}{2}$ — the $x$ -co-ordinate of the midpoint (halfway across).
$\frac{y_1 + y_2}{2}$ — the $y$ -co-ordinate of the midpoint (halfway up).
Add the pair of $x$ -values and halve; add the pair of $y$ -values and halve. Two small averages, not the distance formula.

A segment from A to B with its midpoint M marked exactly halfway, labelled as the average of the two endpoints' co-ordinates.

Worked ExampleFinding a midpoint

A footbridge is to be built at the midpoint of the segment joining posts $A(-2, 5)$ and $B(6, 1)$ . Find its co-ordinates.

Step 1 — Average the $x$ -values

$\frac{x_1 + x_2}{2} = \frac{-2 + 6}{2} = \frac{4}{2} = 2$

Step 2 — Average the $y$ -values

$\frac{y_1 + y_2}{2} = \frac{5 + 1}{2} = \frac{6}{2} = 3$

Step 3 — Write the midpoint

$M = (2, 3)$

Both co-ordinates sit between the endpoints ( $2$ is between $-2$ and $6$ ; $3$ is between $1$ and $5$ ), so the answer is reasonable.

Working backwards from a midpoint

Sometimes you know the midpoint and one endpoint, and must find the other endpoint.
The step from the endpoint to the midpoint is repeated to reach the far end: whatever you add to $x_1$ to reach the midpoint, add again.
Or solve $\frac{x_1 + x_2}{2} = m_x$ for the unknown $x_2$ .

Worked ExampleFinding the far endpoint

$M(4, 1)$ is the midpoint of segment $AB$ . If $A = (1, -3)$ , find $B$ .

Step 1 — Set up an equation for each co-ordinate

Using the midpoint formula with $B = (x_2, y_2)$ :

$\frac{1 + x_2}{2} = 4 \qquad \text{and} \qquad \frac{-3 + y_2}{2} = 1$

Step 2 — Solve each equation

$1 + x_2 = 8 \;\Rightarrow\; x_2 = 7$ $-3 + y_2 = 2 \;\Rightarrow\; y_2 = 5$

Step 3 — State the point

$B = (7, 5)$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Find the midpoint of the segment joining $A(1, 4)$ and $B(7, 10)$ .

Merit

$M(0, 2)$ is the midpoint of $PQ$ . If $P = (-5, -1)$ , find $Q$ .

Excellence

A parallelogram has vertices $A(1, 1)$ , $B(5, 2)$ , $C(7, 6)$ and $D(3, 5)$ . Show that its diagonals bisect each other.

The midpoint of a line segment

Covers AS91256

What the midpoint is

The midpoint is the point exactly halfway along the segment joining two points — the same distance from each end.
Its co-ordinates are the average of the two end co-ordinates: average the $x$ -values, then average the $y$ -values.

The midpoint formula

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

$\frac{x_1 + x_2}{2}$ — the $x$ -co-ordinate of the midpoint (halfway across).
$\frac{y_1 + y_2}{2}$ — the $y$ -co-ordinate of the midpoint (halfway up).
Add the pair of $x$ -values and halve; add the pair of $y$ -values and halve. Two small averages, not the distance formula.

A segment from A to B with its midpoint M marked exactly halfway, labelled as the average of the two endpoints' co-ordinates.

Worked ExampleFinding a midpoint

A footbridge is to be built at the midpoint of the segment joining posts $A(-2, 5)$ and $B(6, 1)$ . Find its co-ordinates.

Step 1 — Average the $x$ -values

$\frac{x_1 + x_2}{2} = \frac{-2 + 6}{2} = \frac{4}{2} = 2$

Step 2 — Average the $y$ -values

$\frac{y_1 + y_2}{2} = \frac{5 + 1}{2} = \frac{6}{2} = 3$

Step 3 — Write the midpoint

$M = (2, 3)$

Both co-ordinates sit between the endpoints ( $2$ is between $-2$ and $6$ ; $3$ is between $1$ and $5$ ), so the answer is reasonable.

Working backwards from a midpoint

Sometimes you know the midpoint and one endpoint, and must find the other endpoint.
The step from the endpoint to the midpoint is repeated to reach the far end: whatever you add to $x_1$ to reach the midpoint, add again.
Or solve $\frac{x_1 + x_2}{2} = m_x$ for the unknown $x_2$ .

Worked ExampleFinding the far endpoint

$M(4, 1)$ is the midpoint of segment $AB$ . If $A = (1, -3)$ , find $B$ .

Step 1 — Set up an equation for each co-ordinate

Using the midpoint formula with $B = (x_2, y_2)$ :

$\frac{1 + x_2}{2} = 4 \qquad \text{and} \qquad \frac{-3 + y_2}{2} = 1$

Step 2 — Solve each equation

$1 + x_2 = 8 \;\Rightarrow\; x_2 = 7$ $-3 + y_2 = 2 \;\Rightarrow\; y_2 = 5$

Step 3 — State the point

$B = (7, 5)$

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

Find the midpoint of the segment joining $A(1, 4)$ and $B(7, 10)$ .

Merit

$M(0, 2)$ is the midpoint of $PQ$ . If $P = (-5, -1)$ , find $Q$ .

Excellence

A parallelogram has vertices $A(1, 1)$ , $B(5, 2)$ , $C(7, 6)$ and $D(3, 5)$ . Show that its diagonals bisect each other.

The midpoint of a line segment

What the midpoint is

The midpoint formula

Step 1 — Average the xxx-values

Step 2 — Average the yyy-values

Step 3 — Write the midpoint

Working backwards from a midpoint

Step 1 — Set up an equation for each co-ordinate

Step 2 — Solve each equation

Step 3 — State the point

Test yourself

Practice by grade

The midpoint of a line segment

What the midpoint is

The midpoint formula

Step 1 — Average the xxx-values

Step 2 — Average the yyy-values

Step 3 — Write the midpoint

Working backwards from a midpoint

Step 1 — Set up an equation for each co-ordinate

Step 2 — Solve each equation

Step 3 — State the point

Test yourself

Practice by grade

Step 1 — Average the $x$ -values

Step 2 — Average the $y$ -values

Step 1 — Average the $x$ -values

Step 2 — Average the $y$ -values