Main content

### Course: High school geometry > Unit 6

Lesson 1: Distance and midpoints# 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.

## Want to join the conversation?

- How do you answer a problem using the midpoint formula when you are only given one end point and the midpoint?(35 votes)
- I don't think you need the answer 11 years later, but I'll still answer 😁

If we are given 1 endpoint and the midpoint, then we use the reverse of the midpoint theorem.

Say we have the endpoint as (0,0) and the midpoint as (3,1).

Then we would graph it as: (0,0) --- (3,1) --- (X,Y)

To Solve X:

∴ ½ ( x of point A + what we are solving for) = x of midpoint

= ½ (0 + X) = 3

= 0 + X = 3 x2

∴ X = 6

To Solve Y:

∴ ½ ( y of point A + what we are solving for) = y of midpoint

= ½ (0 + Y) = 1

= 0 + Y = 1 x2

∴ Y = 2

∴ The other endpoint = (6,2)(26 votes)

- Instead of finding the mean, can you find the median?(13 votes)
- With only two data points, the mean, median and mode are all the same.(29 votes)

- 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)- 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!(4 votes)

- What is average actually?I don't understand difference between mean media mode either(5 votes)
- Take, for example, the numbers 1, 2, 3, 4, and 5.

Average: Add all the numbers together and divide them by 5(because there are 5 numbers that we added together).

1+2+3+4+5 = 15

15/5 = 3

The average of these numbers is 3.

"Mean" is basically just another word for the average.

"Median" is the middle number in a list of numbers that are ordered from least to greatest. In our example set of numbers, the median would also be 3.

"Mode" is the most frequent number. For instance, if you have the numbers 1, 2, 3, 4, 4, and 5, the number 4 would be the mode or most frequent number.

See this video:

https://www.khanacademy.org/math/probability/data-distributions-a1/summarizing-center-distributions/v/statistics-intro-mean-median-and-mode

Hope this helps!(6 votes)

- Hey Sal, isn't the midpoint formula (x1 +x2 / 2), (y1 + y2 / 2?)(5 votes)
- This is the formula he uses in the video, but he just simplifies it. x+1/2 is the same thing as the average or mean, and the same goes for the y coordinate.(1 vote)

- How do you know which number to put in first? The formula is (x1+x2)/2 + (y1+y2)/2, but how do you know which number should be put in x1? Could I just put either of the x-values? I understand all of the distance formula except for this part.(5 votes)
- Addition is commutative, so it doesn't matter here.

Generally, these subscripts serve to group an x/y pair as a point and as long as you use the corresponding x/y pair it does not matter which pair is designated 1 or 2.(4 votes)

- Would finding the hypotenuse and then dividing by 2 also work?(4 votes)
- Yeah, but it's essentially just adding an extra step to what you are already doing. plus you would need extra steps to find the x and y coordinates(1 vote)

- How to find Endpoint

Point A- (-7,-9)

Midpoint- (-0.5, -3)

Point B- (?) - (6,3)

How to find x:

1/2 (-7+x)= -0.5

-3.5 + 1/2x= -0.5

+3.5 +3.5

1/2x = 3

1/2 12

*you divide both sides and eliminate one with the x 1/2 and just divide 3 divided by 1/2

x= 6

You find Y by doing the same just put the coordinates of Y

Your answer is (6,3) for Point B(3 votes) - How do u find the answer to questions like: The point (7,−2) is the midpoint of (6,−6) and what point?(1 vote)
- 7=1/2(6+x) => 2(7)=6+x => 14=6+x => 8=x

-2=1/2(-6+y) => 2(-2)=-6+y => -4=-6+y => 2=y

therefore the other point would be (8,2)(4 votes)

- why did he say smack dab?(2 votes)
- smack dab in the middle is an American colloquialism that is used. It means exactly in the middle.(3 votes)

## 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.