In this video, we're going to focus on finding the midpoint between two points. So let's say we have point A, which is 2,6, and point B, which is 4,10. What is the midpoint between these two points?
Find the coordinates of the midpoint. To find the x-coordinate, all you need to do is average the two x-values. Here's the formula that you can use. It's x1 plus x2 divided by... 2. And the y-coordinate is simply the average of the other two y-coordinates.
So if you were to average 2 and 4, what number will you get? The midpoint between 2 and 4 is 3. So for example, if we do x1 plus x2, let's call this x1 and this one x2. If we add 2 and 4 and by 2 this will give us 2 plus 4 6 and 6 divided by 2 is 3 so that will give us a midpoint of 3 now let's do the same thing with the y-values let's call this y1 and y2. The midpoint between 6 and 10 instinctively you can see is 8. So if you were to add 6 and 10 and then divide by 2 you should get 8. 10 plus 6 is 16, 16 divided by 2 is 8. So that's how you can find the midpoint between two points. So in this example it's.
So now it's your turn. Go ahead and find the midpoint between A and B. Let's say A is 3,5 and B is 9,1.
So feel free to pause the video and try this out for the sake of practice. So let's use the same formula. We're going to call this x1, and this is going to be y1.
This is x2, y2. So it's x1 plus x2, that's 3 plus 9, divided by 2. That's going to give us the x. coordinate of the midpoint. Now to find the y coordinate, it's y1, which is 5, plus y2, which is 1, divided by 2. Now 3 plus 9 is 12, and 5 plus 1 is 6. 12 divided by 2 is 6. 6 divided by 2 is 3. And so this is the midpoint between points A and B.
As you can see, 6 is right in the middle between 3 and 9. 6 is the midpoint between 3 and 9. And 3 is the midpoint between 5 and 1. 3 is exactly in the middle of those two numbers. Here's another one that you could try. It's going to have some negative numbers included in it. Find the midpoint between these two ordered pairs, negative 4, 2 and 8, negative 6. So let's use the same formula, x1 plus x2 divided by 2, y1 plus y2 divided by 2. So let's call this x1, y1. This is going to be x2, and that's y2.
x1 is negative 4, x2 is positive 8. y1 is 2, y2 is negative 6. Negative 4 plus 8, that's equal to positive 4. And 2 plus negative 6, or simply 2 minus 6, that's negative 4. 4 divided by 2 is 2. Negative 4 divided by 2 is negative 2. So, this is the coordinates of the midpoint. It's 2, negative 2. Go ahead and try this one. Find a midpoint between 5, 3 and 8, negative 7. So x1 is 5, x2 is 8, y1 is 3, y2 is negative 7. Now 5 plus 8, that's 13. 3 plus negative 7, that's negative 7. 4 and we know negative 4 divided by 2 is negative 2 now we can't reduce 13 over 2 so we're just going to leave it as a fraction so this is the answer for that this example Now, other times you may have coordinates that contain fractions. So let's say if point A is 1 over 3 comma 2, and point B is 5 comma negative 1 4. So feel free to tackle this problem.
Pause the video and see if you can find the midpoint between these two coordinates, or between those two points. So let's follow the same procedure. But actually, let's do it one step at a time.
Let's find the midpoint of the x coordinate first. We know it's going to be x1 plus x2 divided by 2. So x1 is 1 over 3. x2 is 5. Now we need to simplify this result. So how can we simplify this complex fraction?
What I would recommend doing is multiplying the numerator and the denominator by 3. 2 times 3 is 6. Now on top, you need to use the distributive property. You got to multiply 3 by 1 third, which the 3's will cancel, leaving behind 1. And then you need to multiply 3 times 5, which is 15. So 1 plus 15 is 16. So you have 16 divided by 6, which you can simplify the fraction. 16 is 8 times 2. 6 is 3 times 2. So we can cancel a 2. And so... So the x coordinate of the midpoint is 8 over 3, which I'm going to put right here.
Now what we need to do next is to find... the midpoint I mean the y-coordinate of the midpoint which we know it's going to be y1 plus y2 divided by 2 so y1 is 2 y2 is negative 1 fourth so we can just simply write minus 1 fourth divided by 2. So let's follow the same process to simplify the complex fraction. So you want to get rid of this fraction on top and notice that it has a denominator of 4. Therefore, you want to multiply the top and the bottom. So the 4's will cancel, leaving behind 1. But there's a negative sign in front of the 1, so it's negative 1. 8 minus 1 is 7. So the second part is 7 over 8. So that's how you could find the midpoint if you're given two points that contain fractions.
Now let's work on another example. That's similar to the last one, but with a little bit more work. So let's say... point A is 1 over 5 comma negative 2 over 3 and point B that's going to be let's say 4 over 3 comma 3 over 4. So I want you to find the midpoint between points A and B. So we have a lot of fractions but the principle of working this problem is the same as the last one.
So go ahead and try it. Take a minute and try this problem. So first, let's find the x coordinate of the midpoint.
Let's average the x values. So the first x value is 1 over 5, and the second x value is 4 over 3. And then we're going to divide it by 2. So we're averaging two fractions. Now, what I need to do is get rid of the 5 and the 3. So I'm going to multiply the top and the bottom by 15. Now, what's 1 5th of 15? 1 5th of 15 is the same as 15 divided by 5, which is 3. Next, we need to multiply 15 by 4 over 3. 15 times 4 is 60. 60 divided by 3 is 20. Or you could divide first.
You could say 15 divided by 3 is 5. 5 times 4 is 20. On the denominator, we have 2 times 15, which is 30. So, 3 plus 20 is 23. So, the x-coordinate of the midpoint is 23 over 30. Now, let's do the same for the y-coordinates. y1 is negative 2 over 3. y2 is 3 over 4. Let's divide it by 2. So we'll need to multiply the top and bottom by 12 to get rid of the denominators of the smaller fractions. So the common denominator is 12. So let's multiply 12 by negative 2 thirds. 12 divided by 3 is 4. 4 times 2 is 8. So 2 thirds of 12 is 8. But let's not forget about the negative sign in front of it. And then 12 times 3 fourths.
12 divided by 4 is 3. 3 times 3 is 9. 2 times 12 is 24. Negative 8 plus 9 is 1. So it's 1 over 24. So this is the midpoint between points A and B.