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Midpoint formula

Learn how to use the midpoint formula to find the midpoint of a line segment on the coordinate plane, or find the endpoint of a line segment given one point and the midpoint. Created by Sal Khan.

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  • female robot grace style avatar for user Abby Morgret
    How do you solve a problem like "The point (0,2) is the midpoint of (2,-3) and what point?"
    (160 votes)
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    • blobby green style avatar for user cstarmer
      I love the formula you came up with, but i would argue that you check your math on this one. Here is the answer I came up with:

      0=1/2(2+x) => 2(0)=2+x => 0=2+x => -2=x

      2=1/2(-3+y) => 2(2)=--3+y => 4=--3+y => 7=y

      therefore the other point would be (-2,7)

      I hope this makes sense :)
      (133 votes)
  • purple pi purple style avatar for user Monty
    How would you work this out in 3D? (x,y,z)
    (63 votes)
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  • piceratops seed style avatar for user newyorkgirl286
    How would you find the midpoint of ordered triples?
    (28 votes)
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  • leaf blue style avatar for user Ismene
    How do you answer a problem using the midpoint formula when you are only given one end point and the midpoint?
    (28 votes)
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  • leafers ultimate style avatar for user Taha
    Instead of finding the mean, can you find the median?
    (13 votes)
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  • blobby green style avatar for user IIPanda.Monium13II
    This helped a lot more! However, I'm still confused. Math isn't my thing. I understand stuff better, visually. I don't understand how the mean works. I used to know what an average was all through out elementary, but when we learned this on Friday, I was suddenly confused.
    When you count the half way point of the x values, you get 4.5. That's the median/mean. When the mid point equation was created, did someone notice that the median was the same as the mean so they came up with that equation to make life easy? If not, then what does "mean" actually mean. I know how to get it, i know it's the average .. but what is the average,central number or mean of a set of numbers supposed to be?
    (11 votes)
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    • starky sapling style avatar for user Growing Brain
      Simply defined; A median is a number that is between two numbers which is exactly halfway, like for example; the median of 3 and 5 is 4 (There are many numbers between 3 and 5...but 4 is the median because the difference between 3 and 4 is the same as the difference between 4 and 5) So if you take on the number line, the median of 3 and 5 must be equidistant from both 3 and 5. The formula for finding out the median is the sum of those two numbers divided by two. [ie. (a+b)/2, where a and b are numbers for whom you want to find the median] Here's how it works; Suppose you have a line segment on the number line with start point 3 and end point 5,the midpoint of the segment is 4. We know that point 4 is equidistant from both ends of the segment, but in general we can also find out the length of our segment and divide it by two to find the length between the median and each of the end points...our line segments length is 2 units, (since it starts at 3 and ends at 5) and if we wanted to find the midpoint of the segment, you can simply divide its length into halves and add the value to 3 (just try it on paper!) If we use the formula for median; then its (3+5)/2, we also get 4, but why this works?
      The difference between 3 and 5 is 2, so both numbers can be represented as 3 and (3+2), so in our previous way, we divided the distance by two and added it to the start point, and that's exactly what the formula does...we have added 3 and (3+2) together and divided by two, that is [(3+3)+(2)]/2...notice that the difference between the numbers is getting divided by two and also 3 is repeated two times in the numerator!
      (1 vote)
  • blobby green style avatar for user sd120
    Hey Sal, isn't the midpoint formula (x1 +x2 / 2), (y1 + y2 / 2?)
    (6 votes)
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  • blobby green style avatar for user Brian Babu
    What is average actually?I don't understand difference between mean media mode either
    (5 votes)
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  • aqualine ultimate style avatar for user ProfessionalMind
    Would finding the hypotenuse and then dividing by 2 also work?
    (4 votes)
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  • aqualine tree style avatar for user Safiya Kalil
    How do u find the answer to questions like: The point (7,−2) is the midpoint of (6,−6) and what point?
    (1 vote)
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Video transcript

Let's say I have the point 3 comma negative 4. So that would be 1, 2, 3, and then down 4. 1, 2, 3, 4. So that's 3 comma negative 4. And I also had the point 6 comma 1. So 1, 2, 3, 4, 5, 6 comma 1. So just like that. 6 comma 1. In the last video, we figured out that we could just use the Pythagorean theorem if we wanted to figure out the distance between these two points. We just drew a triangle there and realized that this was the hypotenuse. In this video, we're going to try to figure out what is the coordinate of the point that is exactly halfway between this point and that point? So this right here is kind of the distance, the line that connects them. Now what is the coordinate of the point that is exactly halfway in between the two? What is this coordinate right here? It's something comma something. And to do that-- let me draw it really big here. Because I think you're going to find out that it's actually pretty straightforward. At first it seems like a really tough problem. Gee, let me use the distance formula with some variables. But you're going to see, it's actually one of the simplest things you'll learn in algebra and geometry. So let's say that this is my triangle right there. This right here is the point 6 comma 1. This down here is the point 3 comma negative 4. And we're looking for the point that is smack dab in between those two points. What are its coordinates? It seems very hard at first. But it's easy when you think about it in terms of just the x and the y coordinates. What's this guy's x-coordinate going to be? This line out here represents x is equal to 6. This over here-- let me do it in a little darker color-- this over here represents x is equal to 6. This over here represents x is equal to 3. What will this guy's x-coordinate be? Well, his x-coordinate is going to be smack dab in between the two x-coordinates. This is x is equal to 3, this is x is equal to 6. He's going to be right in between. This distance is going to be equal to that distance. His x-coordinate is going to be right in between the 3 and the 6. So what do we call the number that's right in between the 3 and the 6? Well we could call that the midpoint, or we could call it the mean, or the average, or however you want to talk about it. We just want to know, what's the average of 3 and 6? So to figure out this point, the point halfway between 3 and 6, you literally just figure out, 3 plus 6 over 2. Which is equal to 4.5. So this x-coordinate is going to be 4.5. Let me draw that on this graph. 1, 2, 3, 4.5. And you see, it's smack dab in between. That's its x-coordinate. Now, by the exact same logic, this guy's y-coordinate is going to be smack dab between y is equal to negative 4 and y is equal to 1. So it's going to be right in between those. So this is the x right there. The y-coordinate is going to be right in between y is equal to negative 4 and y is equal to 1. So you just take the average. 1 plus negative 4 over 2. That's equal to negative 3 over 2 or you could say negative 1.5. So you go down 1.5. It is literally right there. So just like that. You literally take the average of the x's, take the average of the y's, or maybe I should say the mean to be a little bit more specific. A mean of only two points. And you will get the midpoint of those two points. The point that's equidistant from both of them. It's the midpoint of the line that connects them. So the coordinates are 4.5 comma negative 1.5. Let's do a couple more of these. These, actually, you're going to find are very, very straightforward. But just to visualize it, let me graph it. Let's say I have the point 4, negative 5. So 1, 2, 3, 4. And then go down 5. 1, 2, 3, 4, 5. So that's 4, negative 5. And I have the point 8 comma 2. So 1, 2, 3, 4, 5, 6, 7, 8 comma 2. 8 comma 2. So what is the coordinate of the midpoint of these two points? The point that is smack dab in between them? Well, we just average the x's, average the y's. So the midpoint is going to be-- the x values are 8 and 4. It's going to be 8 plus 4 over 2. And the y value is going to be-- well, we have a 2 and a negative 5. So you get 2 plus negative 5 over 2. And what is this equal to? This is 12 over 2, which is 6 comma 2 minus 5 is negative 3. Negative 3 over 2 is negative 1.5. So that right there is the midpoint. You literally just average the x's and average the y's, or find their means. So let's graph it, just to make sure it looks like midpoint. 6, negative 5. 1, 2, 3, 4, 5, 6. Negative 1.5. Negative 1, negative 1.5. Yep, looks pretty good. It looks like it's equidistant from this point and that point up there. Now that's all you have to remember. Average the x, or take the mean of the x, or find the x that's right in between the two. Average the y's. You've got the midpoint. What I'm going to show you now is what's in many textbooks. They'll write, oh, if I have the point x1 y1, and then I have the point-- actually, I'll just stick it in yellow. It's kind of painful to switch colors all the time-- and then I have the point x2 y2, many books will give you something called the midpoint formula. Which once again, I think is kind of silly to memorize. Just remember, you just average. Find the x in between, find the y in between. So midpoint formula. What they'll really say is the midpoint-- so maybe we'll say the midpoint x-- or maybe I'll call it this way. I'm just making up notation. The x midpoint and the y midpoint is going to be equal to-- and they'll give you this formula. x1 plus x2 over 2, and then y1 plus y2 over 2. And it looks like something you have to memorize. But all you have to say is, look. That's just the average, or the mean, of these two numbers. I'm adding the two together, dividing by two, adding these two together, dividing by two. And then I get the midpoint. That's all the midpoint formula is.