Using Slope to Partition Segments

An error occurred trying to load this video.

Try refreshing the page, or contact customer support.

Coming up next: How to Find the Slope of a Parallel Line

You're on a roll. Keep up the good work!

Take Quiz Watch Next Lesson
Your next lesson will play in 10 seconds
  • 0:04 Partitioning a Line Segment
  • 0:52 Using Slope
  • 2:34 Example
  • 4:26 Lesson Summary
Save Save Save

Want to watch this again later?

Log in or sign up to add this lesson to a Custom Course.

Log in or Sign up

Speed Speed

Recommended Lessons and Courses for You

Lesson Transcript
Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

Partitioning a line segment involves breaking the line segment up in a particular way. This lesson will show how to use the slope of a line segment to find a point that partitions the line segment into a given ratio.

Partitioning a Line Segment

At the circus, Mike the magnificent is walking the tight rope. It takes him 10 equal size steps to get across the rope. He takes seven steps flawlessly, then wobbles a bit, and quickly takes the last three steps to land safely on the end platform.

The point where Mike wobbles partitions the rope (line segment) into the ratio 7/3.

It just so happens that Mike just performed a mathematical feat called partitioning a line segment. Partitioning a directed line segment, AB, into a ratio a/b involves dividing the line segment into a + b equal parts and finding a point that is a equal parts from A and b equal parts from B.

Partitioning a line segment with ratio a/b

Consider Mike again. The point at which he wobbled on the tight rope (a line segment of 10 equal parts) is seven equal parts from the start and three equal parts from the end, so the point at which Mike wobbled partitioned the line segment into the ratio 7/3.

Using Slope

Partitioning a directed line segment seems simple enough, but what if we are given the two endpoints of a directed line segment, and want to find the point that partitions the line segment into the ratio a/b?

Thankfully, we can do this fairly easily using parts of the slope of the line segment. The slope of the line segment with endpoints (x1, y1) and (x2, y2) gives us the rate at which y is changing with respect to x, and we can find it using the slope formula:

Slope = Rise/Run = (Change in y)/(Change in x) = (y2 - y1)/(x2 - x1)

If we are given a line segment AB, where A = (x1, y1) and B = (x2, y2), and we want to partition it into the ratio a/b, then we want to find a point P that falls a equal parts from point A and b equal parts from point B on the line segment. We can do this using the following steps:

  1. Determine the ratio, call it c, comparing a to the entire length of the line segment using the formula c = a/(a + b). This ratio gives the fraction of the way that P is from A to B.
  2. Find the rise (y2 - y1) and run (x2 - x1) of the slope of the line segment.
  3. Add c⋅(run) to the x1, and add c⋅(rise) to y1. This takes point A and moves it a/(a + b) of the way to point B, which is exactly the point P that we want.

These steps also give way to a nice easy formula for P:

P = (x1 + c(x2 - x1), y1 + c(y2 - y1))


Hmmm…that seems to make sense, but don't you think an example will make things even more clear?


Suppose we have a directed line segment AB, where A = (1,2) and B = (8,7), and we want to partition it with the ratio 3/5. In other words, we want to find a point, P, that is three equal parts from A and is five equal parts from B. Let's take it through our steps, and then we'll verify our answer with our formula.

To unlock this lesson you must be a Member.
Create your account

Register to view this lesson

Are you a student or a teacher?

Unlock Your Education

See for yourself why 30 million people use

Become a member and start learning now.
Become a Member  Back
What teachers are saying about
Try it risk-free for 30 days

Earning College Credit

Did you know… We have over 200 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Transferring credit to the school of your choice

Not sure what college you want to attend yet? has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

Create an account to start this course today
Try it risk-free for 30 days!
Create an account