Lesson Explainer: Partitioning a Line Segment on the Coordinate Plane Mathematics

In this explainer, we will learn how to find the coordinates of a point that divides a line segment on the coordinate plane with a ratio using the section formula.

Let us first review some terminology.

Definition: Line Segment

A line segment is a part of a line bounded by two distinct endpoints.

We can represent the line segment between two distinct points, 𝐴 and 𝐵, using the notation 𝐴𝐵. 𝐴𝐵 contains all the points on the straight line between 𝐴 and 𝐵.

To help us understand this definition, we can consider a line segment 𝐴𝐵 drawn on the coordinate plane with endpoints 𝐴(−4,5) and 𝐵(2,−3).

The midpoint of a line segment is the middle point of the segment, the point that is equidistant between the two endpoints. We can find the coordinates of the midpoint of 𝐴𝐵 by halving each of the horizontal and vertical distances between 𝐴 and 𝐵.

Recap: The Midpoint of a Line Segment

We can find the midpoint, 𝑀, of a line segment between (𝑥,𝑦) and (𝑥,𝑦) using 𝑀=𝑥+𝑥2,𝑦+𝑦2.

We will now look at a variety of questions on dividing, or partitioning, line segments in a number of different ways.

Example 1: Dividing a Line Segment into Four Equal Parts

The coordinates of 𝐴 and 𝐵 are (1,9) and (9,9) respectively. Determine the coordinates of the points that divide 𝐴𝐵 into four equal parts.

Answer

We can begin by sketching the line segment 𝐴𝐵 and showing the points that divide it into 4 equal parts. We can define these points as 𝑃, 𝑀, and 𝑄.

As 𝐴𝐵 is divided into four equal parts, we can approach this question by firstly finding the midpoint, 𝑀, of 𝐴𝐵 and then finding the midpoints of 𝐴𝑀 and 𝑀𝐵.

We recall that the midpoint, 𝑀, of a line segment between coordinates (𝑥,𝑦) and (𝑥,𝑦) is given by 𝑀=𝑥+𝑥2,𝑦+𝑦2.

To find the midpoint, 𝑀, of 𝐴𝐵, we substitute the coordinates of 𝐴(1,9) for the (𝑥,𝑦) values and the coordinates of 𝐵(9,9) for the (𝑥,𝑦) values, giving 𝑀=1+92,9+92=102,182=(5,9). Thus, the coordinates of 𝑀 are (5,9).

Next, we find the midpoint, 𝑃, of 𝐴𝑀. Substituting the coordinates 𝐴(1,9) and 𝑀(5,9) for the (𝑥,𝑦), and (𝑥,𝑦) values, respectively, gives 𝑃=1+52,9+92=62,182=(3,9).

Finally, we find the midpoint, 𝑄, of 𝑀𝐵. Using the coordinates 𝑀(5,9) and 𝐵(9,9) gives 𝑃=5+92,9+92=142,182=(7,9).

Thus, we have found the coordinates of 𝑃, 𝑀, and 𝑄, which divide 𝐴𝐵 into 4 equal parts, as (3,9),(5,9),(7,9).

We will now look an example of how a line segment that has been partitioned by a point can be written in terms of a ratio.

Example 2: Finding the Ratio by Which a Point Divides a Line Segment

Fill in the blank: If 𝐶∈𝐴𝐵 and 𝐴𝐵=3𝐶𝐵, then 𝐶 divides 𝐵𝐴 by the ratio .

  1. 2∶1
  2. 1∶2
  3. 1∶3
  4. 3∶1

Answer

We consider that we have a line segment 𝐴𝐵. Somewhere along this line segment will be point 𝐶.

Since we need to take into consideration the direction of movement from 𝐴 to 𝐵, we use the vector 𝐴𝐵.

The movement from point 𝐶 to point 𝐵 is the vector 𝐶𝐵.

The magnitudes of vectors 𝐴𝐵 and 𝐶𝐵 are their lengths. We are given that 𝐴𝐵=3𝐶𝐵; therefore, we can write that lengthoflengthof𝐴𝐵=3×𝐶𝐵.

We can divide 𝐴𝐵 into 3 equal pieces.

However, we need to establish at which of these points 𝐶 will lie. If 𝐶 is closer to 𝐴 than 𝐵, then the length of 𝐶𝐵 would be two-thirds the length of 𝐴𝐵.

Thus, it would not be true that 𝐴𝐵=3𝐶𝐵. Therefore, 𝐶 must be at the point that is closer to 𝐵 than 𝐴.

In this way, 𝐶𝐵=13𝐴𝐵 and 𝐴𝐵=3𝐶𝐵.

To find the ratio, as 𝐴𝐵 is divided into 3 parts, there will be 2 parts of the total in 𝐴𝐶 and 1 part in 𝐶𝐵.

We could write that 𝐶 divides 𝐴𝐵 in the ratio 2∶1. However, we were asked how 𝐶 divides 𝐵𝐴; therefore, the solution is the ratio given in answer option B: 1∶2.

We will now investigate how we can find the coordinates of a point on a line segment that splits the line into a given ratio.

Let us begin with a line segment 𝐴𝐵, with distinct points 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦). We then wish to find the coordinates of the point, 𝑃, which splits 𝐴𝐵 in the ratio 𝑚∶𝑛.

As 𝑃 is on 𝐴𝐵, vector 𝐴𝑃 has the same direction as 𝐴𝐵. The ratio of the magnitudes of 𝐴𝑃∶𝑃𝐵 is 𝑚∶𝑛.

The length of 𝐴𝑃 can be found by the magnitude of 𝐴𝑃, written as ‖‖𝐴𝑃‖‖. Expressing this in terms of 𝑚 and 𝑛, as a proportion of the length of 𝐴𝐵, we have ‖‖𝐴𝑃‖‖=𝑚𝑚+𝑛×‖‖𝐴𝐵‖‖.

For example, if the length of 𝐴𝑃 is 3 units and the length of 𝑃𝐵 is 2 units, we could write that ‖‖𝐴𝑃‖‖=33+2×‖‖𝐴𝐵‖‖=35×‖‖𝐴𝐵‖‖.

We can now find the coordinates of 𝑃 by considering the length of 𝐴𝑃 in the context of the coordinate plane. We can consider the 𝑥-coordinate and 𝑦-coordinate separately.

𝐴 has an 𝑥-coordinate of 𝑥, and the horizontal displacement from 𝐴 to 𝐵 above can be rewritten in terms of the 𝑥-values as 𝑥−𝑥.

The 𝑥-coordinate of 𝑃 will be the 𝑥-coordinate of 𝐴 plus the horizontal displacement from 𝐴 to 𝑃.

We know that 𝐴𝑃=𝑚𝑚+𝑛𝐴𝐵 and 𝐴𝐵=(𝑥−𝑥,𝑦−𝑦); therefore, the horizontal displacement from 𝐴 to 𝑃 is 𝑚𝑚+𝑛(𝑥−𝑥).

Thus, we can write the 𝑥-coordinate of 𝑃 as 𝑥=𝑥+𝑚𝑚+𝑛(𝑥−𝑥).

In order to simplify this equation, we can write 𝑥 as a fraction by multiplying the numerator and denominator by (𝑚+𝑛): 𝑥=(𝑚+𝑛)𝑥𝑚+𝑛+𝑚(𝑥−𝑥)𝑚+𝑛.

We can prepare to add these fractions, and distribute over the parentheses, giving 𝑥=𝑚𝑥+𝑛𝑥+𝑚𝑥−𝑚𝑥𝑚+𝑛.

Simplifying, we have 𝑥=𝑚𝑥+𝑛𝑥+𝑚𝑥−𝑚𝑥𝑚+𝑛𝑥=𝑚𝑥+𝑛𝑥𝑚+𝑛.

Now, we have a simplified expression for the 𝑥-coordinate of 𝑃.

In the same way, we can write the value of the 𝑦-coordinate of 𝑃 as 𝑦=𝑦+𝑚𝑚+𝑛(𝑦−𝑦).

Simplifying as before, we would obtain that 𝑦=𝑚𝑦+𝑛𝑦𝑚+𝑛.

This allows us to write the following formula.

Theorem: The Section Formula

If we have distinct points 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) and the point 𝑃∈𝐴𝐵 divides 𝐴𝐵 such that 𝐴𝑃∶𝑃𝐵=𝑚∶𝑛, then 𝑃 has the coordinates 𝑃=𝑚𝑥+𝑛𝑥𝑚+𝑛,𝑚𝑦+𝑛𝑦𝑚+𝑛.

We will now see how we can apply this formula in a few example questions.

Example 3: Finding the Coordinate That Divides a Line Segment Internally

If the coordinates of 𝐴 and 𝐵 are (5,5) and (−1,−4), respectively, find the coordinates of point 𝐶 that divides 𝐴𝐵 internally by the ratio 2∶1.

Answer

We can sketch this directed line segment 𝐴𝐵 as shown.

We can apply the following formula to find point 𝐶 that divides 𝐴𝐵 internally in the ratio 2∶1. This means that 𝐶∈𝐴𝐵 and the ratio will be given as 𝐴𝐶∶𝐶𝐵=2∶1.

We can then apply the formula to partition a line segment in a given ratio.

If 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) and point 𝑃 divides 𝐴𝐵 such that 𝐴𝑃∶𝑃𝐵=𝑚∶𝑛, then 𝑃 has the coordinates 𝑃=𝑚𝑥+𝑛𝑥𝑚+𝑛,𝑚𝑦+𝑛𝑦𝑚+𝑛.

For our problem, 𝐴 has coordinates (5,5) and 𝐵 has coordinates (−1,−4). We can substitute these coordinates into the formula for (𝑥,𝑦) and (𝑥,𝑦) respectively.

The ratio values 2∶1 can be substituted for 𝑚 and 𝑛 respectively.

Therefore, we will have the coordinates of 𝐶 as 𝐶=2(−1)+1(5)2+1,2(−4)+1(5)2+1.

Simplifying, we have 𝐶=33,−33=(1,−1).

Thus, the coordinates of point 𝐶 that divides 𝐴𝐵 internally by the ratio 2∶1 are (1,−1).

We will now see how we can partition a line segment externally in a given ratio.

Consider the coordinates 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦). Point 𝑃=(𝑥,𝑦) lies on ray 𝐴𝐵 (not lying between points 𝐴 and 𝐵), such that the ratio 𝐴𝑃∶𝑃𝐵=𝑚∶𝑛.

Using a similar proof to that of internal partitioning, we have that 𝑃 is at a distance of 𝑚𝑚−𝑛×𝐴𝐵 from point 𝐴.

Therefore, the 𝑥-coordinate of 𝑃 can be found by 𝑥=𝑥+𝑚𝑚−𝑛(𝑥−𝑥)=(𝑚−𝑛)𝑥+𝑚𝑥−𝑚𝑥𝑚−𝑛=𝑚𝑥−𝑛𝑥+𝑚𝑥−𝑚𝑥𝑚−𝑛=𝑚𝑥−𝑛𝑥𝑚−𝑛.

Similarly, the 𝑦-coordinate can be written as 𝑦=𝑚𝑦−𝑛𝑦𝑚−𝑛.

This allows us to write the following formula.

Theorem: The Section Formula with External Division

If we have distinct points 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) and point 𝑃∉𝐴𝐵 divides 𝐴𝐵 such that 𝐴𝑃∶𝑃𝐵=𝑚∶𝑛, then 𝑃 has the coordinates 𝑃=𝑚𝑥−𝑛𝑥𝑚−𝑛,𝑚𝑦−𝑛𝑦𝑚−𝑛.

We will now see how we can apply this formula in the following example.

Example 4: Finding the Coordinates of a Point That Divides a Line Segment Externally into a Given Ratio

If 𝐴(3,−2) and 𝐵(−2,4), find in vector form the coordinates of point 𝐶 that divides 𝐴𝐵 externally in the ratio 4∶3.

Answer

We can begin by sketching the points 𝐴 and 𝐵 and extending the directed line segment 𝐴𝐵 to point 𝐶 that divides 𝐴𝐵 externally. We can write that 𝐴𝐶∶𝐶𝐵=4∶3.

We recall the section formula for external division.

If we have distinct points 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) and point 𝑃∉𝐴𝐵 divides 𝐴𝐵 such that 𝐴𝑃∶𝑃𝐵=𝑚∶𝑛, then 𝑃 has the coordinates 𝑃=𝑚𝑥−𝑛𝑥𝑚−𝑛,𝑚𝑦−𝑛𝑦𝑚−𝑛.

We can substitute the values 𝐴(3,−2) and 𝐵(−2,4) for the (𝑥,𝑦) and (𝑥,𝑦) values, respectively, and the ratio values 4∶3 for 𝑚 and 𝑛 into the section formula to find the coordinates of 𝐶. This gives us 𝐶=4(−2)−3(3)4−3,4(4)−3(−2)4−3=−8−91,16+61=(−17,22).

As we are asked to give our answer in vector form, we can give the position vector of 𝐶 as (−17,22).

We will now see an example of how we can use the section formula to find the ratio in which a line segment is divided.

Example 5: Finding the Ratio by Which the 𝑥-Axis Divides a Line Segment

Given 𝐶(−3,3) and 𝐷(4,−2), then the 𝑥-axis divides 𝐶𝐷 by the ratio .

  1. 3∶5
  2. 5∶3
  3. 2∶3
  4. 3∶2

Answer

We can begin by plotting the coordinates 𝐶 and 𝐷 and sketching the vector 𝐶𝐷.

In order to find how the 𝑥-axis divides 𝐶𝐷, we first need to find the point where 𝐶𝐷 crosses the 𝑥-axis. Given the coordinates of 𝐶 and 𝐷, we can find the equation of 𝐶𝐷, beginning by finding the slope of this line.

The slope, 𝑚, of a line joining two points (𝑥,𝑦) and (𝑥,𝑦) can be found using 𝑚=𝑦−𝑦𝑥−𝑥.

Therefore, the slope between 𝐶(−3,3) and 𝐷(4,−2) is given by 𝑚=−2−34−(−3)=−57=−57.

We can then use the point–slope form of a line such that, given a point (𝑥,𝑦) and the slope, 𝑚, we can write the equation of the line as 𝑦−𝑦=𝑚(𝑥−𝑥).

We can substitute the coordinates of either 𝐶 or 𝐷 into this form, so using 𝐶(−3,3) for the (𝑥,𝑦) values and 𝑚=−57, we have 𝑦−3=−57(𝑥−(−3))𝑦−3=−57(𝑥+3).

We can then multiply both sides by 7 and expand the parentheses on the right-hand side, giving 7𝑦−21=−5(𝑥+3)7𝑦−21=−5𝑥−15.

Rearranging to write this in the general form of the equation of a line, 𝑎𝑥+𝑏𝑦+𝑐=0, we have 5𝑥+7𝑦−6=0.

We recall that a line crosses the 𝑥-axis when 𝑦=0, so substituting this into the equation 5𝑥+7𝑦−6=0 and simplifying gives 5𝑥+7(0)−6=0,5𝑥−6=0,5𝑥=6,𝑥=65.

We have now calculated that the line segment 𝐶𝐷 crosses the 𝑥-axis at the point 65,0.

We now need to find the ratio by which this coordinate, 65,0, divides 𝐶𝐷.

To do this, we can use the section formula for internal division of a line segment. If we have distinct points 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) and point 𝑃∈𝐴𝐵 divides 𝐴𝐵 such that 𝐴𝑃∶𝑃𝐵=𝑚∶𝑛, then 𝑃 has the coordinates 𝑃=𝑚𝑥+𝑛𝑥𝑚+𝑛,𝑚𝑦+𝑛𝑦𝑚+𝑛.

In this question, we know points 𝐶(−3,3) and 𝐷(4,−2) and point 𝑃65,0 that divides 𝐶𝐷. We need to calculate the ratio values of 𝑚 and 𝑛.

Substituting 𝐶(−3,3) and 𝐷(4,−2) for the (𝑥,𝑦) and (𝑥,𝑦) values, respectively, into the section formula, we have 𝑃=𝑚(4)+𝑛(−3)𝑚+𝑛,𝑚(−2)+𝑛(3)𝑚+𝑛=4𝑚−3𝑛𝑚+𝑛,−2𝑚+3𝑛𝑚+𝑛.

We know that 𝑃 has coordinates 65,0, so we can write 65,0=4𝑚−3𝑛𝑚+𝑛,−2𝑚+3𝑛𝑚+𝑛.

Evaluating the 𝑥-coordinates, we have 65=4𝑚−3𝑛𝑚+𝑛.

We can cross multiply and simplify to write an expression for 𝑚 in terms of 𝑛 as 5(4𝑚−3𝑛)=6(𝑚+𝑛)20𝑚−15𝑛=6𝑚+6𝑛14𝑚=21𝑛𝑚𝑛=2114𝑚𝑛=32.

The ratio of 𝐶𝑃∶𝑃𝐷=𝑚∶𝑛, so 𝐶𝑃∶𝑃𝐷=3∶2.

Thus, we can give the answer that the 𝑥-axis divides 𝐶𝐷 in the ratio 3∶2.

As a check of our answer, we could find the distance of 𝐶𝑃 and the distance of 𝑃𝐷 and find the ratio of 𝐶𝑃∶𝑃𝐷 directly.

We recall that the distance formula for finding the distance, 𝑑, between two points (𝑥,𝑦) and (𝑥,𝑦) is given by 𝑑=(𝑥−𝑥)+(𝑦−𝑦).

To find the length of 𝐶𝑃, ‖‖𝐶𝑃‖‖, we substitute the values of 𝐶(−3,3) and 𝑃65,0 for the (𝑥,𝑦) and (𝑥,𝑦) values to give ‖‖𝐶𝑃‖‖=65−(−3)+(0−3)=215+3=66625=3√745.

To find the length of 𝑃𝐷, ‖‖𝑃𝐷‖‖, we substitute 𝑃65,0 and 𝐷(4,−2) for the (𝑥,𝑦) and (𝑥,𝑦) values, which gives ‖‖𝑃𝐷‖‖=4−65+(−2−0)=145+(−2)=29625=2√745.

We can then write the ratio of 𝐶𝑃∶𝑃𝐷 as 𝐶𝑃∶𝑃𝐷=3√745∶2√745.

Multiplying both sides of the ratio by 5, and then dividing by √74, gives 𝐶𝑃∶𝑃𝐷=3√74∶2√74=3∶2.

Thus, we have confirmed our answer, that the 𝑥-axis divides 𝐶𝐷 by the ratio 3∶2.

In the next example, we can see a more complex problem involving the partitioning of a line segment.

Example 6: Solving a Word Problem by Dividing a Line Segment

A bus is traveling from city 𝐴(10,−10) to city 𝐵(−8,8). Its first stop is at 𝐶, which is halfway between the cities. Its second stop is at 𝐷, which is two-thirds of the way from 𝐴 to 𝐵. What are the coordinates of 𝐶 and 𝐷?

Answer

We are given that city 𝐴 has coordinates (10,−10) and city 𝐵 has coordinates (−8,8). Firstly, we need to find the coordinates of city 𝐶, halfway between these.

We can make use of the formula for the midpoint of a line. To find the midpoint, 𝑀, of a line segment between two points (𝑥,𝑦) and (𝑥,𝑦) we can use 𝑀=𝑥+𝑥2,𝑦+𝑦2.

Substituting (𝑥,𝑦)=(10,−10) and (𝑥,𝑦)=(−8,8) into this formula gives the midpoint, 𝐶, as 𝐶=10+(−8)2,−10+82=22,−22=(1,−1).

Next, we need to find the coordinates of 𝐷, which is two-thirds of the way from 𝐴 to 𝐵. The direction, 𝐴 to 𝐵, is important as it indicates the position of 𝐷. 𝐷 will be closer to 𝐵 than 𝐴. We can think of the position by dividing 𝐴𝐵 into 3 equal pieces. We can write the ratio 𝐴𝐷∶𝐷𝐵 as 2∶1.

We can use the formula to partition a line segment in a given ratio. If 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) and point 𝑃 divides 𝐴𝐵 such that 𝐴𝑃∶𝑃𝐵=𝑚∶𝑛, then 𝑃 has the coordinates 𝑃=𝑚𝑥+𝑛𝑥𝑚+𝑛,𝑚𝑦+𝑛𝑦𝑚+𝑛.

In this question, we have 𝐴(10,−10), 𝐵(−8,8), and point 𝐷 that divides 𝐴𝐵 in the ratio 2∶1. Substituting these into the formula to find 𝐷 gives 𝐷=2(−8)+1(10)2+1,2(8)+1(−10)2+1=−63,63=(−2,2).

As a useful check of our answer, we can consider the lengths of 𝐴𝐵 and 𝐴𝐷 by applying the distance formula. To find the distance, 𝑑, between two points (𝑥,𝑦) and (𝑥,𝑦), we calculate 𝑑=(𝑥−𝑥)+(𝑦−𝑦).

To calculate the length of 𝐴𝐷, we can substitute the coordinates 𝐴(10,−10) and 𝐷(−2,2) into the formula to give 𝑑=(−2−10)+(2−(−10))=(−12)+12=√288=12√2.

To calculate the length of 𝐴𝐵, we substitute 𝐴(10,−10) and 𝐵(−8,8) into the distance formula, giving 𝑑=(−8−10)+(8−(−10))=(−18)+18=√648=18√2.

Thus, we can write the ratio of lengths 𝐴𝐷∶𝐴𝐵 as 𝐴𝐷∶𝐴𝐵=12√2∶18√2=12∶18=2∶3.

We were given that 𝐷 is two-thirds of the way from 𝐴 to 𝐵. Therefore, we have confirmed that 𝐷 has the coordinates (−2,2).

We can give the answer that the coordinates of 𝐶 and 𝐷 are (1,−1)(−2,2).and

Key Points

  • The midpoint, 𝑀, of the line segment between (𝑥,𝑦) and (𝑥,𝑦) is given by 𝑀=𝑥+𝑥2,𝑦+𝑦2.
  • If we have distinct points 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) and point 𝑃 divides 𝐴𝐵 internally such that 𝐴𝑃∶𝑃𝐵=𝑚∶𝑛, then 𝑃 has the coordinates 𝑃=𝑚𝑥+𝑛𝑥𝑚+𝑛,𝑚𝑦+𝑛𝑦𝑚+𝑛.
  • When answering problems involving the partitioning of a line segment, we need to be careful to establish the correct order of the ratio. If 𝐴𝐵 is split by point 𝑃 in the ratio 𝑚∶𝑛, then the ratio of 𝐴𝑃∶𝑃𝐵 will be 𝑚∶𝑛. If 𝐵𝐴 is split by point 𝑃 in the ratio 𝑚∶𝑛, then the ratio of 𝐴𝑃∶𝑃𝐵 will instead be 𝑛∶𝑚.
  • If we have distinct points 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦) and point 𝑃∉𝐴𝐵 divides 𝐴𝐵 externally such that 𝐴𝑃∶𝑃𝐵=𝑚∶𝑛, then 𝑃 has the coordinates 𝑃=𝑚𝑥−𝑛𝑥𝑚−𝑛,𝑚𝑦−𝑛𝑦𝑚−𝑛.

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