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How to Find the Midpoint (Mathematics Lesson)

Finding the Midpoint

The midpoint is half way between two points.

We can find the Cartesian coordinates of the midpoint if we know the Cartesian coordinates of the two points, .

Imagine we wanted to find the midpoint between the points with Cartesian coordinates (1, 2) and (5, 4). The image below shows what we mean by the midpoint:

How to Find the Midpoint

Finding the midpoint between two points is easy.

An Example Question

Find the midpoint between the points with Cartesian coordinates (1, 2) and (5, 4).
Step 1
Add the x-coordinates of the two points together. The x-coordinate of (1, 2) is 1. The x-coordinate of (5, 4) is 5.
1 + 5 = 6
Step 2
Divide the answer (6) by 2.
6 ÷ 2 = 3
3 is the x-coordinate of the midpoint.

Step 3
Add the y-coordinates of the two points together. The y-coordinate of (1, 2) is 2. The y-coordinate of (5, 4) is 4.
2 + 4 = 6
Step 4
Divide the answer (6) by 2.
6 ÷ 2 = 3
3 is the y-coordinate of the midpoint.

Step 5
Write down the Cartesian coordinates of the midpoint as a pair of numbers in brackets, separated by a comma.

The x-coordinate (3) found in Step 2 goes on the left.

The y-coordinate (3) found in Step 4 goes on the right.



How to Find the Midpoint Using a Formula

The midpoint between a point at (x1, y1) and a point at (x2, y2) is found at:



The image below shows what we mean by x1, y1, x2 and y2:

A Real Example of How to Find the Midpoint

The slider below gives another example of how to find the midpoint of two points.

Interactive Widget
Here is an interactive widget to help you learn about finding the midpoint.

Interactive Test
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Here's a second test on finding the midpoint.
Here's a third test on finding the midpoint.
Note

What Are Cartesian Coordinates?

Cartesian coordinates are used to describe the position of a point on a graph.

Cartesian coordinates work by measuring how far across and how far up the point is from the origin.

Why Does the Method Work?

The method works because it finds the point in the middle of the two points.

The x-coordinate of the midpoint is just the average of the x-coordinates of the end points.

The y-coordinate of the midpoint is just the average of the y-coordinates of the end points.

Remember, the average of two numbers is found by adding them together and dividing by 2.

The average of two numbers is the number exactly half way between the two numbers.