Midpoint Formula

Written by

Malcolm McKinsey

January 11, 2023

Fact-checked by

Paul Mazzola

Midpoint definition

With any two ordered pairs, a midpoint exists that lies exactly halfway between each ordered pair. This is true in two dimensions (x and y coordinates) and three dimensions (x, y, and z coordinates).

With two dimensions, you have only two endpoints, so the midpoint (or mean) is also the median, making your mathematics very easy.

Midpoint formula

The Midpoint Formula is used to find the exact center point between two defined points in a line segment. Use this formula to calculate the point that bisects a line segment

How to find midpoint

In a coordinate grid, straight line segments can be horizontal (flat, like the horizon, along the X-axis), vertical (straight up and down, along the Y-axis), or diagonal (at a slant). You can only find midpoints of line segments, not lines, since lines have no end.

You need:

• Both endpoints to find the midpoint

• One endpoint and a midpoint to find the other endpoint

How to find midpoint of horizontal line segments

To calculate the midpoint of a horizontal line segment, focus on the x values, add them and divide by two:

The mean and median, and therefore the middle or midpoint of the line, has an x value of 5. The midpoint is (5, 4).

How to find midpoint of vertical line segments

To calculate the length of a vertical line segment, concentrate on the y values:

The midpoint is at (2, 6.5).

How to find midpoint of diagonal line segments

Diagonal line segments are a lot trickier than finding the midpoint of vertical or horizontal line segments. Here is the ideal place for the midpoint formula, which essentially finds the average of the x values and y values:

You see how you are adding the two x values, and then dividing by 2? That finds the average, which is the midpoint, for the x value. Repeat for the y values, and together you have the ordered pair of the midpoint.

Midpoint formula examples

The Midpoint Formula works with line segments in all quadrants. Suppose you had this:

Plug in the endpoints, being careful with the negative numbers:

Then, plug in to find y:

The midpoint is at (−5, −2.5).

How to find the midpoint between two points

Do not be discouraged when your line segment crosses from one quadrant to another. The Midpoint Formula still works. You do have to be careful of your x values and y values, but just plug in the numbers, divide, and you have the midpoint.

Plug in the two endpoints:

Now, let's do the second endpoint:

The midpoint is (0, 0), the origin of the coordinate grid!

Find the coordinates of the midpoint

Sometimes you get very little information, like an endpoint and the midpoint. You are asked to find the other endpoint. You can do this!

Remember that the midpoint is the average of only two sets of numbers. Use that to help you find the missing x value and y values, the second endpoint, $({x}_{2},{y}_{2})$.

You are given (-7, -3) as one endpoint $({x}_{1},{y}_{1})$ and (0, -1) as the midpoint.

You need to find these values:

For x:

Multiply both sides by 2 and then simplify:

Finally, add 7 to both sides:

For y:

Multiply both sides by 2 and then simplify:

Finally, add 3 to both sides:

This yields the endpoint $({x}_{2},{y}_{2})$ of $(7,1)$.

Lesson summary

Now that you worked all the way down to here, you are able to identify the midpoint formula and determine its appropriate use, explain to yourself and others how to find the midpoint of a vertical or horizontal line segment on a coordinate grid.

You are also able to use the Midpoint Formula to calculate the endpoint of a diagonal line given the midpoint and one endpoint and use the same formula to calculate the midpoint of a diagonal line given two endpoints.