Develop and use the midpoint formula

Ontario Grade 10 Mathematics

Analytic Geometry

Find the point halfway between two coordinates—and use it to find a missing endpoint.

A midpoint is the point exactly halfway between two endpoints of a line segment. On a number line, the point halfway between 2 and 8 is 5: it is 3 units from each endpoint. The same idea works on a coordinate grid. Find the halfway value horizontally and the halfway value vertically. In this lesson, you will connect that familiar idea to a formula and practise using it.

What you will learn

Explain what a midpoint represents on a coordinate grid.
Develop the midpoint formula by averaging the coordinates of two endpoints.
Use the formula to find a midpoint or a missing endpoint.

1. Review: halfway on a number line

An endpoint is one of the two ends of a line segment. A coordinate is a number that tells where a point is. On a number line, the midpoint between two numbers is their average: add them, then divide by 2. For example, the midpoint between 2 and 8 is 5 because

\frac{2+8}{2}=5

.

The midpoint is the same distance from both endpoints. Here, 5 is 3 units from 2 and 3 units from 8. This is a useful way to check whether a result makes sense.

On a coordinate grid, each point has two coordinates. The first tells its horizontal position, and the second tells its vertical position. To find a point halfway between two plotted points, find the halfway position in each direction.

\text{average}=\frac{\text{first number}+\text{second number}}{2}

A midpoint is exactly halfway between two endpoints.
For two numbers, the average is their sum divided by 2.
A point on a coordinate grid has a horizontal coordinate and a vertical coordinate.

2. From halfway positions to the midpoint formula

Consider endpoints

A(-4,2)

and

B(6,8)

. Their horizontal coordinates are

-4

and

6

. The horizontal halfway position is their average, or

1

. Their vertical coordinates are

2

and

8

. The vertical halfway position is

5

. So the point halfway between them is

(1,5)

.

The table shows that the two coordinates are averaged separately. The letters

x_1,y_1

name the coordinates of the first endpoint, and

x_2,y_2

name the coordinates of the second endpoint. The subscript numbers just distinguish the first point from the second; they do not mean that a coordinate is squared.

The midpoint formula collects these two averages in one rule. It works whether the coordinates are positive, negative, or zero. The order of the endpoints does not change the midpoint, because adding the same two values gives the same sum.

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

Average the horizontal coordinates with each other.
Average the vertical coordinates with each other.
The first coordinate of the midpoint comes from the two horizontal coordinates; the second comes from the two vertical coordinates.

3. Use the formula when an endpoint is missing

The midpoint formula can also help when you know the midpoint and one endpoint but not the other. For each coordinate, the midpoint is the average of the two endpoint coordinates. You can use that relationship to find the missing value.

For example, if the midpoint’s horizontal coordinate is 4 and one endpoint’s horizontal coordinate is 1, the other horizontal coordinate must be 7. The average of 1 and 7 is 4. Apply the same idea to the vertical coordinates. Keep the horizontal calculation separate from the vertical calculation.

A helpful check is to average the two endpoint coordinates you have found. Each average should equal the corresponding coordinate of the given midpoint. If it does not, revisit the arithmetic or the coordinate pairing.

x_2=2x_M-x_1,\qquad y_2=2y_M-y_1

A known midpoint and one endpoint can determine the other endpoint.
Work with horizontal coordinates as one pair and vertical coordinates as another.
Check a missing endpoint by recalculating the midpoint.

4. Practise and check your result

For independent practice, find the midpoint of

C(3,-5)

and

D(9,1)

. Then find the missing endpoint

Q

if the midpoint of

P(-2,4)

and

Q

is

(3,1)

. Show the horizontal and vertical calculations separately. Check each answer by averaging the two endpoint coordinates.

Before you finish, ask yourself: Did I combine coordinates in the same direction? Did I divide each sum by 2 when finding a midpoint? Did I keep negative signs? These checks catch many common errors.

Practise the same coordinate-by-coordinate process on both questions.
Use averaging to check a midpoint or a missing endpoint.
A midpoint must lie between the endpoints in both coordinate directions.

Coordinate-by-coordinate midpoint calculation

Direction	Endpoint A	Endpoint B	Average: midpoint coordinate
Horizontal	$-4$	$6$	$\frac{-4+6}{2}=1$
Vertical	$2$	$8$	$\frac{2+8}{2}=5$

Worked example

Find a midpoint and use it to recover an endpoint

The endpoints of a segment are

A(-4,2)

and

B(6,8)

. Find the midpoint. Then suppose the midpoint is known to be

(1,5)

and the first endpoint is still

A(-4,2)

. Find the second endpoint.

Average the horizontal coordinates
The horizontal coordinates are $-4$ and $6$ . Add them and divide by 2 to find the halfway horizontal position.
$\frac{-4+6}{2}=1$
Average the vertical coordinates
The vertical coordinates are $2$ and $8$ . Their average gives the halfway vertical position.
$\frac{2+8}{2}=5$
Write the midpoint
Pair the horizontal halfway value with the vertical halfway value. The midpoint is therefore $(1,5)$ .
$(1,5)$
Find the missing horizontal coordinate
For the second part, let the unknown endpoint be $B(x_2,y_2)$ . The midpoint’s horizontal coordinate is 1, so the average of $-4$ and $x_2$ must be 1. Multiply both sides by 2, then add 4 to isolate the unknown coordinate.
$\frac{-4+x_2}{2}=1\quad\Longrightarrow\quad x_2=6$
Find the missing vertical coordinate
The midpoint’s vertical coordinate is 5, so the average of 2 and $y_2$ must be 5. Multiply by 2, then subtract 2 to find the unknown vertical coordinate.
$\frac{2+y_2}{2}=5\quad\Longrightarrow\quad y_2=8$

Answer: The midpoint is

(1,5)

. With endpoint

A(-4,2)

and midpoint

(1,5)

, the other endpoint is

B(6,8)

.

Check: Averaging the endpoint coordinates gives

\frac{-4+6}{2}=1

and

\frac{2+8}{2}=5

, which matches the given midpoint.

Common mistakes and how to avoid them

Averaging a horizontal coordinate with a vertical coordinate.

Correction: Pair the two horizontal coordinates together, then pair the two vertical coordinates together.

Adding the coordinates but forgetting to divide by 2.

Correction: The midpoint is halfway, so each coordinate is the average of the matching endpoint coordinates.

Dropping a negative sign when adding coordinates.

Correction: Keep the sign attached to each coordinate. For example,

-4+6=2

, not 10.

Writing the midpoint coordinates in the wrong order.

Correction: Write the horizontal result first and the vertical result second, as

(x,y)

. Check the order used for the endpoints.

Lesson summary

A midpoint is exactly halfway between two endpoints.
Find its horizontal coordinate by averaging the endpoints’ horizontal coordinates.
Find its vertical coordinate by averaging the endpoints’ vertical coordinates.
If one endpoint is missing, use the fact that the midpoint coordinate is the average of the two endpoint coordinates.
Check the result by averaging the endpoint coordinates again.

Check your understanding

Question 1

What is the midpoint of

(2,-3)

and

(8,5)

?

$(5,1)$
$(10,2)$
$(3,6)$
$(5,-1)$

Show answer and explanation

(5,1)

Average matching coordinates:

\frac{2+8}{2}=5

and

\frac{-3+5}{2}=1

. The midpoint is

(5,1)

.

Question 2

A segment has midpoint

(4,2)

and one endpoint

(1,-2)

. What is the other endpoint?

$(7,6)$
$(5,0)$
$(3,4)$
$(7,2)$

Show answer and explanation

(7,6)

The midpoint coordinate is the average of the endpoint coordinates. The missing horizontal coordinate is 7 because the average of 1 and 7 is 4. The missing vertical coordinate is 6 because the average of

-2

and 6 is 2. The other endpoint is

(7,6)

.

Question 3

Which calculation finds the vertical coordinate of the midpoint of

(3,7)

and

(9,-1)

?

$\frac{7+(-1)}{2}$
$\frac{3+9}{2}$
$\frac{7+9}{2}$
$\frac{3+(-1)}{2}$

Show answer and explanation

\frac{7+(-1)}{2}

The vertical coordinates are the second coordinates, 7 and

-1

. Average those two values to get the midpoint’s vertical coordinate, 3.

Key terms

Endpoint: One of the two ends of a line segment.
Coordinate: A number that gives the position of a point in one direction on a grid.
Midpoint: The point exactly halfway between the endpoints of a line segment.
Average: The sum of a set of numbers divided by how many numbers there are. For two numbers, add them and divide by 2.

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About this lesson and its review

Published by DoAssignment. This reviewed lesson follows Ontario Grade 10 Mathematics (MPM2D), study topic G6. It is a study resource, not an official curriculum publication.

Before publication, content is checked for structure, mathematical or chemical notation, calculations, course boundaries, and readability. Errors can still occur, so corrections are welcomed.

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