1.5.4 Midpoint of a line segment

What you'll learn

What the midpoint of a line segment means.
How to derive and use the midpoint formula.
How to work accurately with negative and fractional coordinates.
How midpoint reasoning can solve coordinate geometry problems.

Coordinates and line segments

A point in two-dimensional coordinate geometry is written as an ordered pair $(x, y)$ .

The $x$ -coordinate gives the horizontal position.
The $y$ -coordinate gives the vertical position.

The order matters: $(3, 7)$ and $(7, 3)$ are different points.

A line segment is the finite, straight part of a line joining two endpoints. For example, the segment joining $A(x_1,y_1)$ to $B(x_2,y_2)$ is often written as $A B$ .

Definition

Midpoint

The midpoint of a line segment is the point exactly halfway between its two endpoints. It divides the segment into two equal lengths.

If $M$ is the midpoint of $A B$ , then $A M = M B$ .

Finding a halfway value

Before working with two coordinates at once, consider values on a number line.

The value halfway between two numbers is their mean, found by adding the numbers and dividing by 2:

value2.\text{halfway value}=\frac{\text{first value}+\text{second value}}{2}.

This works with positive values, negative values and fractions.

Example

Finding a halfway value

Find the value halfway between $- 7$ and $5$ .

Add the two endpoint values:
$- 7 + 5 = - 2.$
Divide the result by 2:
$−22=−1.\frac{-2}{2}=-1.$
Check the distances: from $- 7$ to $- 1$ is 6 units, and from $- 1$ to $5$ is also 6 units. Therefore, the halfway value is $- 1$ .

Tip

Quick mental check

The halfway value must lie between the two endpoint values. If your answer lies outside them, check your addition and division.

Building the midpoint formula

Suppose the endpoints of a line segment are

A(x1,y1)andB(x2,y2).A(x_1,y_1) \quad \text{and} \quad B(x_2,y_2).

To reach the midpoint:

move halfway from $x_1$ to $x_2$ horizontally;
move halfway from $y_1$ to $y_2$ vertically.

Therefore, the midpoint's $x$ -coordinate is the mean of $x_1$ and $x_2$ , while its $y$ -coordinate is the mean of $y_1$ and $y_2$ .

Key Idea

The midpoint formula

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

M(x1+x22,y1+y22).M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).

Average the $x$ -coordinates together and average the $y$ -coordinates together.

The subscripts simply identify which point each coordinate belongs to. They are labels, not powers.

Example

Finding a midpoint

Find the midpoint of the segment joining $A (2, 7)$ and $B (10, 3)$ .

Average the $x$ -coordinates:
$2+102=6.\frac{2+10}{2}=6.$
Average the $y$ -coordinates:
$7+32=5.\frac{7+3}{2}=5.$
Combine the two results in coordinate order. The midpoint is $M (6, 5)$ .

Common Mistake

Mixing coordinate types

Do not average an $x$ -coordinate with a $y$ -coordinate. Keep the calculation organised as “ $x$ with $x$ , then $y$ with $y$ ”.

Negative coordinates

The formula does not change when coordinates are negative. However, brackets help you handle the signs correctly.

For instance, if an $x$ -coordinate is $- 8$ , write the numerator carefully before simplifying.

Example

Using negative coordinates

Find the midpoint of the segment joining $P (- 8, 5)$ and $Q (4, - 9)$ .

Average the $x$ -coordinates:
$−8+42=−42=−2.\frac{-8+4}{2}=\frac{-4}{2}=-2.$
Average the $y$ -coordinates:
$5+(−9)2=−42=−2.\frac{5+(-9)}{2}=\frac{-4}{2}=-2.$
Pair these coordinates in the correct order. The midpoint is $(- 2, - 2)$ .

Common Mistake

Losing a negative sign

When one coordinate is negative, substitute it with its sign included. For example, use $5 + (- 9)$ , not $5 + 9$ .

Fractional midpoints

A midpoint does not have to have integer coordinates. If the sum of a pair of coordinates is odd, dividing by 2 produces a value ending in one half.

Unless a decimal is requested, an exact fraction is usually the clearest form.

Example

Finding a fractional midpoint

Find the midpoint of the segment joining $C (- 3, 4)$ and $D (8, 11)$ .

Average the $x$ -coordinates:
$−3+82=52.\frac{-3+8}{2}=\frac{5}{2}.$
Average the $y$ -coordinates:
$4+112=152.\frac{4+11}{2}=\frac{15}{2}.$
Write the exact coordinate pair:
$M(52,152).M\left(\frac{5}{2},\frac{15}{2}\right).$

Why the formula works

The change in the $x$ -coordinate from $A$ to $B$ is $x_2-x_1$ . Half of this change is

x2−x12.\frac{x_2-x_1}{2}.

Starting at $x_1$ and adding this half-change gives

x1+x2−x12=2x1+x2−x12=x1+x22.x_1+\frac{x_2-x_1}{2} =\frac{2x_1+x_2-x_1}{2} =\frac{x_1+x_2}{2}.

The same reasoning applies to the $y$ -coordinate. This shows that the formula genuinely moves halfway along the segment; it is not just a rule to memorise.

Midpoints in problem solving

A question may not directly say “use the midpoint formula”. Instead, midpoint reasoning may be hidden inside information about a shape.

The diagonals of a parallelogram bisect each other. To bisect means to divide into two equal parts. Therefore, the diagonals of a parallelogram have the same midpoint.

Example

Checking a parallelogram

The points are $A (1, 2)$ , $B (7, 4)$ , $C (9, 10)$ and $D (3, 8)$ . Check whether the diagonals $A C$ and $B D$ have the same midpoint.

Find the midpoint of $A C$ :
$(1+92,2+102)=(5,6).\left(\frac{1+9}{2},\frac{2+10}{2}\right)=(5,6).$
Find the midpoint of $B D$ :
$(7+32,4+82)=(5,6).\left(\frac{7+3}{2},\frac{4+8}{2}\right)=(5,6).$
The diagonals have the same midpoint, so they bisect each other. This confirms that the four points form a parallelogram in the given order.

Tip

Checking your result

Your midpoint's $x$ -coordinate should lie between the endpoints' $x$ -coordinates, and its $y$ -coordinate should lie between their $y$ -coordinates. This provides a quick check even when the segment slopes downwards.

Finding a missing endpoint

You can also reverse the midpoint formula. If you know one endpoint and the midpoint, form an equation for each coordinate.

For example, if $A(x_1,y_1)$ and midpoint $M(m_x,m_y)$ are known, then the other endpoint $B(x_2,y_2)$ satisfies

mx=x1+x22andmy=y1+y22.m_x=\frac{x_1+x_2}{2} \quad \text{and} \quad m_y=\frac{y_1+y_2}{2}.

Example

Finding the other endpoint

The midpoint of $A B$ is $M (4, - 1)$ and one endpoint is $A (- 2, 5)$ . Find $B (x, y)$ .

Use the midpoint's $x$ -coordinate:
$−2+x2=4⇒−2+x=8⇒x=10.\frac{-2+x}{2}=4 \Rightarrow -2+x=8 \Rightarrow x=10.$
Use the midpoint's $y$ -coordinate:
$5+y2=−1⇒5+y=−2⇒y=−7.\frac{5+y}{2}=-1 \Rightarrow 5+y=-2 \Rightarrow y=-7.$
Therefore, the other endpoint is $B (10, - 7)$ . Checking gives the midpoint
$(−2+102,5+(−7)2)=(4,−1).\left(\frac{-2+10}{2},\frac{5+(-7)}{2}\right)=(4,-1).$

Exam technique

In the exam

Write the midpoint formula or clearly show that you are averaging corresponding coordinates.
Substitute negative coordinates with their signs included, using brackets where helpful.
Give your final answer as an ordered pair and keep fractions exact unless a decimal is requested.
Check that each midpoint coordinate lies between the corresponding endpoint coordinates.

Self review

Check yourself

What is the midpoint of the segment joining $(- 6, 3)$ and $(8, 11)$ ?
How can you quickly detect that you have mixed an $x$ -coordinate with a $y$ -coordinate?
A segment has midpoint $(2, 5)$ and one endpoint $(- 3, 9)$ . What is the other endpoint?

What you'll learn

What the midpoint of a line segment means.
How to derive and use the midpoint formula.
How to work accurately with negative and fractional coordinates.
How midpoint reasoning can solve coordinate geometry problems.

Coordinates and line segments

A point in two-dimensional coordinate geometry is written as an ordered pair $(x, y)$ .

The $x$ -coordinate gives the horizontal position.
The $y$ -coordinate gives the vertical position.

The order matters: $(3, 7)$ and $(7, 3)$ are different points.

A line segment is the finite, straight part of a line joining two endpoints. For example, the segment joining $A(x_1,y_1)$ to $B(x_2,y_2)$ is often written as $A B$ .

Definition

Midpoint

The midpoint of a line segment is the point exactly halfway between its two endpoints. It divides the segment into two equal lengths.

If $M$ is the midpoint of $A B$ , then $A M = M B$ .

Finding a halfway value

Before working with two coordinates at once, consider values on a number line.

The value halfway between two numbers is their mean, found by adding the numbers and dividing by 2:

value2.\text{halfway value}=\frac{\text{first value}+\text{second value}}{2}.

This works with positive values, negative values and fractions.

Example

Finding a halfway value

Find the value halfway between $- 7$ and $5$ .

Add the two endpoint values:
$- 7 + 5 = - 2.$
Divide the result by 2:
$−22=−1.\frac{-2}{2}=-1.$
Check the distances: from $- 7$ to $- 1$ is 6 units, and from $- 1$ to $5$ is also 6 units. Therefore, the halfway value is $- 1$ .

Tip

Quick mental check

The halfway value must lie between the two endpoint values. If your answer lies outside them, check your addition and division.

Building the midpoint formula

Suppose the endpoints of a line segment are

A(x1,y1)andB(x2,y2).A(x_1,y_1) \quad \text{and} \quad B(x_2,y_2).

To reach the midpoint:

move halfway from $x_1$ to $x_2$ horizontally;
move halfway from $y_1$ to $y_2$ vertically.

Therefore, the midpoint's $x$ -coordinate is the mean of $x_1$ and $x_2$ , while its $y$ -coordinate is the mean of $y_1$ and $y_2$ .

Key Idea

The midpoint formula

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

M(x1+x22,y1+y22).M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right).

Average the $x$ -coordinates together and average the $y$ -coordinates together.

The subscripts simply identify which point each coordinate belongs to. They are labels, not powers.

Example

Finding a midpoint

Find the midpoint of the segment joining $A (2, 7)$ and $B (10, 3)$ .

Average the $x$ -coordinates:
$2+102=6.\frac{2+10}{2}=6.$
Average the $y$ -coordinates:
$7+32=5.\frac{7+3}{2}=5.$
Combine the two results in coordinate order. The midpoint is $M (6, 5)$ .

Common Mistake

Mixing coordinate types

Do not average an $x$ -coordinate with a $y$ -coordinate. Keep the calculation organised as “ $x$ with $x$ , then $y$ with $y$ ”.

Negative coordinates

The formula does not change when coordinates are negative. However, brackets help you handle the signs correctly.

For instance, if an $x$ -coordinate is $- 8$ , write the numerator carefully before simplifying.

Example

Using negative coordinates

Find the midpoint of the segment joining $P (- 8, 5)$ and $Q (4, - 9)$ .

Average the $x$ -coordinates:
$−8+42=−42=−2.\frac{-8+4}{2}=\frac{-4}{2}=-2.$
Average the $y$ -coordinates:
$5+(−9)2=−42=−2.\frac{5+(-9)}{2}=\frac{-4}{2}=-2.$
Pair these coordinates in the correct order. The midpoint is $(- 2, - 2)$ .

Common Mistake

Losing a negative sign

When one coordinate is negative, substitute it with its sign included. For example, use $5 + (- 9)$ , not $5 + 9$ .

Fractional midpoints

A midpoint does not have to have integer coordinates. If the sum of a pair of coordinates is odd, dividing by 2 produces a value ending in one half.

Unless a decimal is requested, an exact fraction is usually the clearest form.

Example

Finding a fractional midpoint

Find the midpoint of the segment joining $C (- 3, 4)$ and $D (8, 11)$ .

Average the $x$ -coordinates:
$−3+82=52.\frac{-3+8}{2}=\frac{5}{2}.$
Average the $y$ -coordinates:
$4+112=152.\frac{4+11}{2}=\frac{15}{2}.$
Write the exact coordinate pair:
$M(52,152).M\left(\frac{5}{2},\frac{15}{2}\right).$

Why the formula works

The change in the $x$ -coordinate from $A$ to $B$ is $x_2-x_1$ . Half of this change is

x2−x12.\frac{x_2-x_1}{2}.

Starting at $x_1$ and adding this half-change gives

x1+x2−x12=2x1+x2−x12=x1+x22.x_1+\frac{x_2-x_1}{2} =\frac{2x_1+x_2-x_1}{2} =\frac{x_1+x_2}{2}.

The same reasoning applies to the $y$ -coordinate. This shows that the formula genuinely moves halfway along the segment; it is not just a rule to memorise.

Midpoints in problem solving

A question may not directly say “use the midpoint formula”. Instead, midpoint reasoning may be hidden inside information about a shape.

The diagonals of a parallelogram bisect each other. To bisect means to divide into two equal parts. Therefore, the diagonals of a parallelogram have the same midpoint.

Example

Checking a parallelogram

The points are $A (1, 2)$ , $B (7, 4)$ , $C (9, 10)$ and $D (3, 8)$ . Check whether the diagonals $A C$ and $B D$ have the same midpoint.

Find the midpoint of $A C$ :
$(1+92,2+102)=(5,6).\left(\frac{1+9}{2},\frac{2+10}{2}\right)=(5,6).$
Find the midpoint of $B D$ :
$(7+32,4+82)=(5,6).\left(\frac{7+3}{2},\frac{4+8}{2}\right)=(5,6).$
The diagonals have the same midpoint, so they bisect each other. This confirms that the four points form a parallelogram in the given order.

Tip

Checking your result

Finding a missing endpoint

You can also reverse the midpoint formula. If you know one endpoint and the midpoint, form an equation for each coordinate.

For example, if $A(x_1,y_1)$ and midpoint $M(m_x,m_y)$ are known, then the other endpoint $B(x_2,y_2)$ satisfies

mx=x1+x22andmy=y1+y22.m_x=\frac{x_1+x_2}{2} \quad \text{and} \quad m_y=\frac{y_1+y_2}{2}.

Example

Finding the other endpoint

The midpoint of $A B$ is $M (4, - 1)$ and one endpoint is $A (- 2, 5)$ . Find $B (x, y)$ .

Use the midpoint's $x$ -coordinate:
$−2+x2=4⇒−2+x=8⇒x=10.\frac{-2+x}{2}=4 \Rightarrow -2+x=8 \Rightarrow x=10.$
Use the midpoint's $y$ -coordinate:
$5+y2=−1⇒5+y=−2⇒y=−7.\frac{5+y}{2}=-1 \Rightarrow 5+y=-2 \Rightarrow y=-7.$
Therefore, the other endpoint is $B (10, - 7)$ . Checking gives the midpoint
$(−2+102,5+(−7)2)=(4,−1).\left(\frac{-2+10}{2},\frac{5+(-7)}{2}\right)=(4,-1).$

Exam technique

In the exam

Write the midpoint formula or clearly show that you are averaging corresponding coordinates.
Substitute negative coordinates with their signs included, using brackets where helpful.
Give your final answer as an ordered pair and keep fractions exact unless a decimal is requested.
Check that each midpoint coordinate lies between the corresponding endpoint coordinates.

Self review

Check yourself

What is the midpoint of the segment joining $(- 6, 3)$ and $(8, 11)$ ?
How can you quickly detect that you have mixed an $x$ -coordinate with a $y$ -coordinate?
A segment has midpoint $(2, 5)$ and one endpoint $(- 3, 9)$ . What is the other endpoint?

What you'll learn

Coordinates and line segments

Finding a halfway value

Building the midpoint formula

Negative coordinates

Fractional midpoints

Why the formula works

Midpoints in problem solving

Finding a missing endpoint

Review 1.5.4 Midpoint of a line segment by teaching Genie

1.5.4 Midpoint of a line segment

What you'll learn

Coordinates and line segments

Finding a halfway value

Building the midpoint formula

Negative coordinates

Fractional midpoints

Why the formula works

Midpoints in problem solving

Finding a missing endpoint

Review 1.5.4 Midpoint of a line segment by teaching Genie

1.5.4 Midpoint of a line segment

What you'll learn

Coordinates and line segments

Finding a halfway value

Building the midpoint formula

Negative coordinates

Fractional midpoints

Why the formula works

Midpoints in problem solving

Finding a missing endpoint

Review 1.5.4 Midpoint of a line segment by teaching Genie

1.5.4 Midpoint of a line segment Revision Guide

1.5.4 Midpoint of a line segment

What you'll learn

Coordinates and line segments

Finding a halfway value

Building the midpoint formula

Negative coordinates

Fractional midpoints

Why the formula works

Midpoints in problem solving

Finding a missing endpoint

Review 1.5.4 Midpoint of a line segment by teaching Genie

1.5.4 Midpoint of a line segment Revision Guide