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Explorations in Core Math - Geometry: Online Textbook Help13 chapters | 127 lessons

Instructor:
*Laura Pennington*

Laura has taught collegiate mathematics and holds a master's degree in pure mathematics.

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 a line segment into a given ratio.

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.

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*.

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

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 (*x*1, *y*1) and (*x*2, *y*2) 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*) = (*y*2 - *y*1)/(*x*2 - *x*1)

If we are given a line segment *AB*, where *A* = (*x*1, *y*1) and *B* = (*x*2, *y*2), 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:

- 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*. - Find the rise (
*y*2 -*y*1) and run (*x*2 -*x*1) of the slope of the line segment. - Add
*c*⋅(run) to the*x*1, and add*c*⋅(rise) to*y*1. 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* = (*x*1 + *c*(*x*2 - *x*1), *y*1 + *c*(*y*2 - *y*1))

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 3 equal parts from *A* and is 5 equal parts from *B*. Let's take it through our steps, and then we'll verify our answer with our formula.

The first thing we want to do is find the ratio *c* that tells us what fraction of the way *P* is from *A* to *B*. To do this, we use our formula *c* = *a*/ (*a* + *b*), where *a* = 3 and *b* = 5.

*c*= 3/(3 + 5) = 3/8

Okay, so we know *P* will be 3/8 of the way from *A* to *B*. Now we want to find the rise and the run of the slope of the line segment. Again, we can use our formulas with the points (1,2) and (8,7).

- Run =
*x*2 -*x*1 = 8 - 1 = 7 - Rise =
*y*2 -*y*1 = 7 - 2 = 5

Alright, one more step! We need to add *c*⋅(run) to *x*1, and add *c*⋅(rise) to *y*1. This adds 3/8 of the change in *x* from *A* to *B* to the *x*-coordinate of *A* and 3/8 of the change in *y* from *A* to *B* to the *y*-coordinate of *A*, giving our desired point *P* that is 3/8 of the way from A to B and partitions the line into the ratio 3/5.

*x*1 +*c*⋅(run) = 1 + (3/8)(7) = 3.625*y*1 +*c*⋅(rise) = 2 + (3/8)(5) = 3.875

This gives that the point *P* is (3.625, 3.875). Let's see if our formula gives the same thing:

It does! Let's take a look at the point *P* on *AB* on a graph.

Sure enough, the point *P* = (3.625, 3.875) partitions *AB* into the ratio 3/5.

**Partitioning a 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*.

When finding a point, *P*, to partition a line segment, *AB*, into the ratio *a*/*b*, we first find a ratio *c* = *a* / (*a* + *b*).

The slope of a line segment with endpoints (*x*1, *y*1) and (*x*2, *y*2) is given by the formula rise/run, where

- rise =
*y*2 -*y*1 - run =
*x*2 -*x*1

Given a line segment *AB* and a partitioning ratio *a*/*b*, we can find the point *P* that partitions the line segment appropriately using a formula that involves parts of the slope of the line segment. That is,

*P*= (*x*1 +*c*(*x*2 -*x*1),*y*1 +*c*(*y*2 -*y*1))

This formula makes partitioning a line segment a cinch, so it is definitely something to keep in your math toolbox for future use!

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Explorations in Core Math - Geometry: Online Textbook Help13 chapters | 127 lessons

- Parallel, Perpendicular and Transverse Lines 6:06
- Angles Formed by a Transversal 7:40
- Types of Angles: Vertical, Corresponding, Alternate Interior & Others 10:28
- Proving Theorems About Perpendicular Lines
- Parallel Lines: How to Prove Lines Are Parallel 6:55
- Perpendicular Bisector Theorem: Proof and Example 6:41
- Using Slope to Partition Segments
- How to Find the Slope of a Perpendicular Line
- Graphs of Parallel and Perpendicular Lines in Linear Equations 6:07
- Go to Explorations in Core Math Geometry Chapter 3: Parallel and Perpendicular Lines

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