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Midpoint Formula

3 min readdecember 13, 2021


Geometry 🔷

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How to Find the Midpoint of Two Points: The Midpoint Formula

Imagine that you have two points on a coordinate plane and your teacher asks you to find the midpoint. This task requires a bit more work, because unlike a singular plane where you just add two points and divide by two, you have to find the center point of two planes, the x and y planes. Luckily there is a formula, the midpoint formula, and it looks like this:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-OJmGX5VOe7Pf.PNG?alt=media&token=3a19f0dd-4cab-4509-8f13-6f28c43cffca

Components

Let’s identify the components of the formula.
  • (xm,ym)=midpoint coordinates
  • (x1,y1)=first set of coordinates
  • (x2,y2)=second set of coordinates

Practice Problems

Question 1

Find the midpoint between these two points.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-V9oNoXhMDTFG.PNG?alt=media&token=bfb5cb78-8ebb-44a8-aada-3de21f6c6e96
The first step is identifying the components of the formula. It doesn’t matter which coordinate point you choose for the first and second set of coordinates. 
(x1,y1)= (4,3)
(x2,y2)= (12,7) 
Now, let’s plug this into the formula.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-xzRKZEHsogLv.PNG?alt=media&token=7f1b0434-61ef-40f4-8a5a-f0b1920d4617
Then just solve like normal:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-YKLZYeGwVwS7.PNG?alt=media&token=c87822a2-14f6-4745-97af-0d98e39e370c
The midpoint is (8,5) which makes sense if you look at it on the graph.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-N9M9TdAuAN3u.PNG?alt=media&token=e26115fd-cec8-4028-9beb-7c07d291194c
This coordinate is right in the middle. 
Remember how it was mentioned that it didn’t matter which coordinate point you choose for the first and second set of coordinates? Let’s see why by switching the first and second set of coordinates. 
(x1,y1)= (12,7)
(x2,y2)= (4,3)
Now, let’s plug this into the formula.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-DwlsByUeYQTA.PNG?alt=media&token=2b9b02f7-3d03-432e-bcd9-86dbb552c593
Bam! Same answer🎊

Question 2

Find the midpoint between these two points.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-GMfyyuszy434.PNG?alt=media&token=37ff9b25-e116-4372-af4d-33a211ab5745
The first step is identifying the components of the formula. 
(x1,y1)= (-4,2)
(x2,y2)= (10,9) 
Now let’s plug this into the formula:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-1XsnqJU9sS5f.PNG?alt=media&token=f4444782-3db3-4c49-8409-a277554aa935
Then, just solve like normal:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-s1EhCgLvhxvG.PNG?alt=media&token=ea3b1ace-a7a5-4875-9a1f-5d89fc30d261
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-s2HB3bgDXIrB.PNG?alt=media&token=f8b4512a-612d-49b7-87b3-2a561f0c3676
The midpoint is (3,5.5).
Sometimes your answer will be a decimal, and that is ok.

Question 3

Find the missing point using the following information:
(10,6)=midpoint coordinates
(4,5)=first set of coordinates
(x,7)=second set of coordinates 
In this practice you must find a missing point, however, you can use the midpoint formula to find the missing point. Just plug in all the information you have in the formula. 
(xm,ym)=(10,6) 
(x1,y1)= (4,5)
(x2,y2)= (x,7)
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-YmnymUOOa0pr.PNG?alt=media&token=2b6510e4-7307-400a-972e-3caf92424532
Then just solve like normal:
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-veixTo7w544M.PNG?alt=media&token=055d9fcc-f7b2-43a0-a70c-5d78dfea7c32
Pause! Here, we can eliminate a few things. We know the midpoint for the y is 6, so we can remove this segment to make things easier.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-WD87KlxUhWl5.PNG?alt=media&token=502370a5-09bb-4a48-a0a3-d9f08f7b1f71
Now just solve for x. First multiply by 2 on both sides to get rid of the fraction.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-EUDoPuICZMEr.PNG?alt=media&token=39c2769a-f1fa-4f85-931b-e87b0f7b916b
Subtract 4 from both sides.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-KEG9crh0ox1W.PNG?alt=media&token=c82fcb5c-fcd7-43ea-b743-3598561a2b7f
And just like that, we found the missing point.

Question 4

Find the missing coordinate point using the following information:
(8,3)=midpoint coordinates
(3,1)=first set of coordinates
(x,y)=second set of coordinates
In this practice you must find a missing coordinate, however, you can use the midpoint formula to find the missing coordinate. Just plug in all the information you have in the formula.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-vMbcs5qtWTEA.PNG?alt=media&token=aefdd39f-7471-4d81-a677-64a3eb56a2be
The first thing you can do is separate the x and y segments.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-Z2QzlxKbdLq0.PNG?alt=media&token=ca81d8d6-1261-4a0a-a257-9b1c5fdfe411
Now just solve for x. First, multiply by 2 on both sides to get rid of the fraction. Let’s start with the x part.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-pgd6iEF7d6QZ.PNG?alt=media&token=c6971fef-16b4-425e-8c28-725ac4c1cca1
Subtract 3 from both sides.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-4KZygcL4InCw.PNG?alt=media&token=61f01ed5-db70-4e35-94bb-b401589fa752
Now let’s find the y point. First multiply by 2 on both sides to get rid of the fraction.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-cp8cy0hKDcyc.PNG?alt=media&token=fadd0eeb-15ed-4f3d-9d22-d09adf734cf9
Subtract 1 from both sides.
https://firebasestorage.googleapis.com/v0/b/fiveable-92889.appspot.com/o/images%2F-XNdlM9jCtwzP.PNG?alt=media&token=d10fdd29-3a95-4be8-8aff-d41a65d0f211
There you go! We found that the missing coordinate is (13,5).

Conclusion

We did it! Midpoints aren’t as difficult as they sound, and we also learned that the midpoint formula can solve other problems.
🤝Connect with other students studying the midpoint formula with Hours.

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