Sometimes you need to find the point that is exactly midway between two other points. For instance, you might need to find a line that bisects (divides into two equal halves) a given line segment. This middle point is called the "midpoint". The concept doesn't come up often, but the Formula is quite simple and obvious, so you should easily be able to remember it for later.

Think about it this way: If you are given two numbers, you can find the number exactly between them by averaging them, by adding them together and dividing by two. For example, the number exactly halfway between 5 and 10 is:

Content Continues Below

The Midpoint Formula works exactly the same way. If you need to find the point that is exactly halfway between two given points, just average the *x*-values and the *y*-values.

#### Find the midpoint

*P*between (−1, 2) and (3, −6).

First, I apply the Midpoint Formula; then, I'll simplify:

So the answer is *P* = (1, −2).

Technically, the Midpoint Formula is the following:

The Midpoint Formula: The midpoint of two points, (*x*_{1}, *y*_{1}) and (*x*_{2}, *y*_{2}) is the point *M* found by the following formula:

Affiliate

But as long as you remember that you're averaging the two points' *x*- and *y*-values, you'll do fine. It won't matter which point you pick to be the "first" point you plug in. Just make sure that you're adding an *x* to an *x*, and a *y* to a *y*.

#### Find the midpoint

*P*between (6.4, 3) and (−10.7, 4).

I'll apply the Midpoint Formula, and simplify:

So the answer is *P* = (−2.15, 3.5).

Content Continues Below

#### Find the value of

*p*so that (−2, 2.5) is the midpoint between (*p*, 2) and (−1, 3).

I'll apply the Midpoint Formula:

The *y*-coordinates already match. This reduces the problem to needing to compare the *x*-coordinates, "equating" them (that is, setting them equal, because they must be the same) and solving the resulting equation to figure out what *p* is. This will give me the value necessary for making the *x*-values match. So:

So the answer is *p* = −3.

Let's do some more examples....

URL: https://www.purplemath.com/modules/midpoint.htm

© 2022 Purplemath, Inc. All right reserved. Web Design by