Midpoint Formula

What is the Midpoint Formula

If you need to find exactly halfway or midway between two points, then it may not be surprising to learn that you will need to use the midpoint formula. The name says it all.

Let’s say you have two coordinates, and you need to find the halfway point in between these two coordinates. That is where the midpoint formula comes in handy.

However, do you know what coordinates are? Look at the numbers below. These are a pair of coordinates. The first coordinate shows where you can plot a point on a graph. Then you can plot the second coordinate on a graph. In most cases, students will draw a straight line through the points to create a line. At times, you may need to find out where the halfway point is on that line. You would use this midpoint formula.

(8,9) (2,4)

So, what do these coordinates mean? The first number, 8, is on the x-axis. The second number, 9, is on the y-axis. The same can be said for the coordinates, 2 and 4. The number 2 is on the x-axis, and the 4 is on the y-axis. Many times, the x and y are labeled over the coordinates as shown below.

$$\begin{array}{ccccc}x&y&&x&y\\\boldsymbol(\mathbf8\boldsymbol,&\mathbf9\boldsymbol)&&\boldsymbol(\mathbf2\boldsymbol,&\mathbf4\boldsymbol)\end{array}$$

In order to use the midpoint formula, you will need to label the coordinates. They have already been labeled with X and Y, but you will need to label the first set of coordinates and the second set.

So, the first number in the first coordinate should be $$x_1$$ and the second number in the first coordinate should be $$y_1$$. Then for the second coordinate, the first number is $$x_2$$ and the second number is $$y_2$$. This will make it easier to plug in these numbers into the formula.

$$\begin{array}{ccccc}x_1&y_1&&x_2&y_2\\\boldsymbol(\mathbf8\boldsymbol,&\mathbf9\boldsymbol)&&\boldsymbol(\mathbf2\boldsymbol,&\mathbf4\boldsymbol)\end{array}$$

The formula for finding the midpoint is below. You will need to use both sets of coordinates to find the midpoint coordinate.

$$X_m Y_m=\left(\frac{x_{1+}x_2}2,\;\frac{y_{1+}y_2}2\right)$$

Now, what do all of these letters and numbers mean? The $$X_m$$ and $$Y_m$$ represent the X and Y coordinate for the midpoint or the halfway point. $$x_1$$ and $$y_1$$ represent the first coordinate. The $$x_2$$ and $$y_2$$ represent the second coordinate.

How Do You Use the Midpoint Formula

Once you have determined what each item represents, you can start plugging in the numbers. However, let’s review what everything represents in the formula. Look at the formula for the midpoint below and see what each symbol represents as well as where you need to go to get the numbers to plug in.

The midpoint coordinate $$(X_{m\;}\;Y_m)$$ is your answer. It is the midpoint. If you plotted both coordinates, and then you plotted the midpoint coordinate, it should look like it is exactly in the middle of those two points.

You will add both x-axis numbers together and then divide by two. This will be the x-axis midpoint number. Then you will add both y-axis numbers together and then divide by two. This will be the y-axis midpoint number.

Fact 1: It is easy to confuse which numbers to plug in because the numbers of the first coordinate go into two different places. They are not placed right next to each other in the formula.

EXAMPLE
It’s time to do this example with the coordinates (8, 9) and (2, 4). Look at how we work the problem below.
$$X_mY_m=\left(\frac{8+2}2,\;\frac{9+4}2\right)$$

$$X_mY_m=\frac{10}2,\;\frac{13}2$$

$$X_mY_m=5,\;6.5$$

To find the midpoint, plug in the numbers from the coordinates for $$x_1$$ and $$y_1$$ and $$x_2$$ and $$y_2$$. Then divide each by 2. For these two coordinates, the $$x_1$$ number is 8 and the $$y_1$$ number is 2.

So, 8 + 2 equals to 10. Then when you divide by 2, your answer is 5. So, the first number in the midpoint for the x-axis is 5. The $$x_2$$ number is 9 and the $$y_2$$ number is 4. When you add these two numbers together, the sum is 13. When you divide 13 by 2, the answer is a decimal, 6.5.

It is fine to have a decimal for a number in your coordinate. So, 5 and 6.5 are the X and Y coordinates for the midpoint for coordinates 8, 9 and 2, 4.

FACT: This formula may appear similar if you try to find the average for a set of numbers. You add a certain amount of numbers together. Then you divide that sum by the amount of numbers you added. The midpoint is halfway between the set of numbers, so the average is considered halfway between a set of numbers too.

When Will I Need to Use the Midpoint Formula?

Obviously, you will not use the midpoint formula during your daily life. Instead, it will be during certain algebraic problems. The most obvious problems will be when the problem will ask in a straightforward way.

It will ask you to find the midpoint or middle of the two points. Then two coordinates are provided.

Another type of problem where you will use the midpoint formula is when a line is given between two points, and you must intersect the line halfway through it. You would then need to use the midpoint formula.

Finally, the last way you will use the midpoint formula is when a shape is given. It will most likely be shown on a graph with an x-axis and y-axis. That is so you will have the coordinates needed to find the midpoint using the midpoint formula.

Coordinates would be needed, so a problem would not simply have just a shape. For example, a triangle could be drawn on a grid, and the problem would ask you to find the center of the triangle or center of one of the lines.

Also, finding the midpoint of a line may be just one step of a problem. You may have to find the midpoint in order to solve the rest of the problem.

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