Unit 1 geometry basics homework 4 partitioning a segment answer key

Pushpanjali Kholia, Laura Pennington

Pushpanjali Kholia
Pushpanjali is an Honours Graduate in Mathematics stream & also holds a Master's Degree in Mathematics from the same college i.e. Kirori Mal College, University of Delhi. She has also done Masters in Public Administration. She has taught higher grades Mathematics for over 5 years and has a good experience in online teaching.
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Instructor Laura Pennington
Laura received her Master's degree in Pure Mathematics from Michigan State University, and her Bachelor's degree in Mathematics from Grand Valley State University. She has 20 years of experience teaching collegiate mathematics at various institutions.
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Learn about partitioning a line segment using slope. Understand how to use the partitioning a line segment formula to partition a line segment given a ratio. Updated: 03/02/2022

A line that has two endpoints is known as a line segment. Dividing or separating the line segment into different measures is known as Partitioning a line segment.

A line segment can be partitioned into many smaller segments. These smaller segments are compared as ratios. To understand the same, let AB be any line segment and partitioning AB into ratio {eq}a:b {/eq} means dividing it into a + b equal parts and then finding point C which is a equal parts from A and b equal parts from B.

Partitioning a line segment

For example: If a line segment AB is divided into a ratio of {eq}2:3 {/eq}, it shows that the line segment can be divided into 5 equal sections. Hence, there are 5 parts in total, in which 2 parts will be before the desired point and 3 parts will be after the desired point.

Partitioning

Thus, partitioning of a line segment means dividing the line segment in the given ratio. The two numbers in the ratio must add up together to equal the total number of partitions of the line segment.

Example: Label point C, such that it divides the line segment AB into a ratio of {eq}3:2 {/eq}

Partitioning AB in the ratio 3:2

Example: Label point X, such that it divides the line segment PQ into a ratio of {eq}4:1 {/eq}.

Partitioning PQ in the ratio 4:1

Note: To understand the meaning of partitioning a directed line segment into a given particular ratio, it is important to understand the basic difference with the help of a simple example that partitioning a line segment in a ratio of {eq}1:2 {/eq} is not the same as finding half the length of the same line segment. The ratio of {eq}1:2 {/eq} means that there will be 3 parts in total, thus dividing the line segment into 3 equal parts, in which 1 part will be before the desired point of partition and 2 parts will be after the desired point of partition.

Ratio of Partitioning Directed Line Segments

Let AB be any line segment with end points{eq}A\left(x_{1},y_{1}\right) {/eq} and {eq}B\left(x_{2},y_{2}\right) {/eq}.

Now to find a point, P, to partition a line segment, AB, into the ratio {eq}a:b {/eq}, such that a equal parts from A and b equal parts from B on the line segment, first find the ratio c which is given by,

{eq}c = \frac{a}{a+b} {/eq}

This ratio gives the fraction of the total length that P is from A to B.

Ratio of partitioning

Example: Partitioning AB into a ratio {eq}2:3 {/eq} where A is {eq}(2,1) {/eq} and B is {eq}(3,5) {/eq}.

Ratio of AB

{eq}A = (2, 1) {/eq} {eq}B = (3, 5) {/eq} {eq}a = 2 {/eq} {eq}b = 3 {/eq}

So, ratio {eq}c = \frac{a}{a+b} = \frac{2}{2+3} = \frac{2}{5} {/eq}

In the above-mentioned case, the ratio of 2:3 shows that the line segment will be divided into 5 equal parts. If P is the partition point, then it will be positioned at 2 of the equal parts away from A and 3 of the equal parts away from B. Hence, P will be closer to A than it will be to B.

So, ratio of directed segment is given by {eq}c = \frac{a}{a+b} {/eq}

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.

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Partitioning a Line Segment Formula

Slope is defined as the steepness of a line which can be calculated by finding the ratio of the "vertical change" to the "horizontal change" between two distinct points on a line. In partitioning of the line segment, using parts of the slope of the line segment, the point that partitions the line segment into the ratio {eq}a:b {/eq} can be found.

In other words, the slope of the line segment having end points at {eq}A\left(x_{1},y_{1}\right) {/eq} and {eq}B\left(x_{2},y_{2}\right) {/eq} gives the rate at which y is changing with respect to x.

So, let AB be any line segment with end points {eq}A\left(x_{1},y_{1}\right) {/eq} and {eq}B\left(x_{2},y_{2}\right) {/eq} and let the partition ratio be {eq}a:b {/eq}. To find the partitioning directed line segment formula, first, find

the change in the {eq}x {/eq} coordinates is {eq}\left(x_{2} - x_{1}\right) {/eq} which is the run.

Similarly, the change in the {eq}y {/eq} coordinates is {eq}\left(y_{2} - y_{1}\right) {/eq} which is the rise.

Slope = Rise/Run = (Change in y)/(Change in x) = {eq}\left(y_{2} - y_{1}\right)/\left(x_{2} - x_{1}\right) {/eq}

Rise and Run in the segment

Add c to {eq}x_{1} {/eq}, and add c to {eq}y_{1} {/eq}. This takes point A and moves it {eq}\frac{a}{a+b} {/eq} to point B, which is point P. From this, the formula for P that is thepartitioning a line segment formula is given by

{eq}P =(x_{1} + c(x_{2} - x_{1}), y_{1} + c(y_{2} - y_{1})) {/eq}.

Example: Let AB be any line segment where {eq}A = (3,4) {/eq} and {eq}B = (5,7) {/eq}, find the rise and the run of the slope of AB.

Slope = Rise/Run = {eq}\left(y_{2} - y_{1}\right)/\left(x_{2} - x_{1}\right) {/eq}

= {eq}(7- 4)/(5-3) {/eq}

= {eq}3/2 {/eq}

How to Partition a Line Segment

In order to find out a partition on a number line, it is required to first find out the total distance between the two given points and accordingly find out the partition of it by multiplying with the required ratio of partition of the segment.

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:

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 (y2 - y1) and run (x2 - x1) of the slope of the line segment.
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?

Example

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.

Partitioning a Line Segment

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

Partitioning a line segment with ratio a/b

Using Slope

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

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 (y2 - y1) and run (x2 - x1) of the slope of the line segment.
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?

Example

How do you partition a segment on a number line?

Thus, the particular point on the number line which represents a partition over the number out of the given length is found.

Example: Find a point X on the number line PQ given by {eq}P(-8) {/eq} and {eq}Q(2) {/eq} such that {eq}PX = \frac{3}{5} PQ {/eq}.

Here, {eq}PQ = |-8-2| = |-10| = 10 {/eq}

{eq}PX = \frac{3}{5} PQ = \frac{3}{5}.10 = 6 {/eq}

Point on number line

Therefore, X point is given by = {eq}-8+ 6 =2 {/eq}

What does partitioning a line segment mean?

Partitioning of a line segment means dividing the line segment in the given ratio. The two numbers in the ratio must add up together to equal the total number of partitions of the line segment.

For example: If a line segment AB is divided into a ratio of {eq}2:3 {/eq}, it shows that the line segment can be divided into 5 equal sections. Hence, there are 5 parts in total, in which 2 parts will be before the desired point and 3 parts will be after the desired point.

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How do you solve a partition segment?

Lesson Summary 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).

What is the formula for partitioning a line segment?

Formula: Position Vector of a Point Partitioning a Line Segment by a Ratio. Let 𝑃 be a point on line segment 𝐴 𝐵 , partitioning it in the ratio 𝑚 ∶ 𝑛 . Then, the position vector  𝑂 𝑃 is given by  𝑂 𝑃 = 𝑚 𝑚 + 𝑛  𝑂 𝐵 + 𝑛 𝑚 + 𝑛  𝑂 𝐴 .

What is segment partitioning?

Partition means to separate or to divide. A line segment can be partitioned into smaller segments which are compared as ratios. Partitions occur on line segments that are referred to as directed segments. A directed segment is a segment that has distance (length) and direction.

What is a partition geometry?

In geometry, space partitioning is the process of dividing a space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions.