Lesson Explainer: Partitioning a Line Segment on the Coordinate Plane | Nagwa Lesson Explainer: Partitioning a Line Segment on the Coordinate Plane | Nagwa

Lesson Explainer: Partitioning a Line Segment on the Coordinate Plane Mathematics • First Year of Secondary School

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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

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.

Vectors can be useful when partitioning line segments in a ratio. Recall that vectors represent direction and magnitude, rather than position on a coordinate plane. Given two distinct points 𝐴 and 𝐵, vector 𝐴𝐵 tells us the relative direction of point 𝐵 with respect to point 𝐴, as well as the distance between the two points. In particular, 𝐴𝐵 does not have to begin at 𝐴 or end at 𝐵, as long as it has the same direction and magnitude. This flexibility of vectors is an advantage when we work with geometric problems such as partitioning a line.

Let us consider how to identify coordinate points. If point 𝑃 partitions 𝐴𝐵 in the ratio 𝑚∶𝑛, this means that point 𝑃 lies on the line segment 𝐴𝐵 and the ratio of the magnitudes of the vectors satisfies ‖‖𝐴𝑃‖‖∶‖‖𝑃𝐵‖‖=𝑚∶𝑛.

In other words, if ‖‖𝐴𝑃‖‖ is 𝑚 length units, ‖‖𝐴𝐵‖‖ would be equal to 𝑚+𝑛 length units, which leads to ‖‖𝐴𝑃‖‖=𝑚𝑚+𝑛‖‖𝐴𝐵‖‖.

In general, it would be difficult to use only the equation above to find the coordinates of the partitioning point 𝑃. However, this equation does not contain the information that 𝑃 lies on the line segment 𝐴𝐵. In particular, this means that 𝐴𝑃 has the same direction as 𝐴𝐵. Recall that two vectors have the same direction if one vector is obtained by multiplying the other vector by some positive constant. Considering the equation above, we can see that this positive constant is given by 𝑚𝑚+𝑛. This leads to 𝐴𝑃=𝑚𝑚+𝑛𝐴𝐵.

We can use this property to find the coordinates of a point that partitions a directed line segment in a given ratio. To achieve this, we first write 𝐴𝑃 and 𝐴𝐵 each as a difference of two position vectors: 𝐴𝑃=𝑂𝑃−𝑂𝐴,𝐴𝐵=𝑂𝐵−𝑂𝐴.

Substituting these expressions into the original formula leads to 𝑂𝑃−𝑂𝐴=𝑚𝑚+𝑛𝑂𝐵−𝑂𝐴.

Rearranging the equation so that 𝑂𝑃 is the subject, we obtain 𝑂𝑃=𝑚𝑚+𝑛𝑂𝐵−𝑂𝐴+𝑂𝐴𝑂𝑃=𝑚𝑚+𝑛𝑂𝐵+−𝑚𝑚+𝑛+1𝑂𝐴𝑂𝑃=𝑚𝑚+𝑛𝑂𝐵+𝑛𝑚+𝑛𝑂𝐴.

Formula: Position Vector of a Point Partitioning a Line Segment by a Ratio

Let 𝑃 be a point on line segment 𝐴𝐵, partitioning it in the ratio 𝑚∶𝑛. Then, the position vector 𝑂𝑃 is given by 𝑂𝑃=𝑚𝑚+𝑛𝑂𝐵+𝑛𝑚+𝑛𝑂𝐴.

Let us see how we can use this formula to obtain an expression for the Cartesian coordinates of the partitioning point. Let us denote the coordinates of the points 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦). Then, we can write the corresponding position vectors 𝑂𝐴=(𝑥,𝑦),𝑂𝐵=(𝑥,𝑦).

Substituting these expressions into the formula above, we obtain 𝑂𝑃=𝑚𝑚+𝑛(𝑥,𝑦)+𝑛𝑚+𝑛(𝑥,𝑦)=𝑚𝑚+𝑛𝑥,𝑚𝑚+𝑛𝑦+𝑛𝑚+𝑛𝑥,𝑛𝑚+𝑛𝑦=𝑚𝑚+𝑛𝑥+𝑛𝑚+𝑛𝑥,𝑚𝑚+𝑛𝑦+𝑛𝑚+𝑛𝑦=𝑚𝑥+𝑛𝑥𝑚+𝑛,𝑚𝑦+𝑛𝑦𝑚+𝑛.

We arrive at 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.

So far, we have observed how to identify the coordinates of a point that divides a line segment in a given ratio. We refer to such problems as internal partitioning problems since the point that we are looking for lies within the line segment.

Let us now consider a different type of problems, known as external division problems. In these problems, the point that divides the line segment does not lie within but rather on an extension of the line segment as shown on the diagram below.

Considering the diagram above, we say that point 𝑃 divides 𝐴𝐵 externally in the ratio 𝑚∶𝑛, where 𝑚>𝑛. We can solve such external division problems by slightly modifying our previous approach to internal partitioning problems. The main difference for this case is that 𝐴𝑃, with magnitude 𝑚 length units, is the larger vector compared to 𝐵𝑃, with magnitude 𝑛 length units. By subtraction, we can see that the length of 𝐴𝐵 is equal to 𝑚−𝑛 units. This leads to ‖‖𝐴𝑃‖‖=𝑚𝑚−𝑛‖‖𝐴𝐵‖‖.

As in the previous context, 𝐴𝑃 and 𝐴𝐵 have the same direction, so we can write 𝐴𝑃=𝑚𝑚−𝑛𝐴𝐵.

As we have done for the internal partitioning problem, we can calculate the formula for the position vector of 𝑃: 𝑂𝑃−𝑂𝐴=𝑚𝑚−𝑛𝑂𝐵−𝑂𝐴𝑂𝑃=𝑚𝑚−𝑛𝑂𝐵−𝑂𝐴+𝑂𝐴𝑂𝑃=𝑚𝑚−𝑛𝑂𝐵+−𝑚𝑚−𝑛+1𝑂𝐴𝑂𝑃=𝑚𝑚−𝑛𝑂𝐵−𝑛𝑚−𝑛𝑂𝐴.

We note that this formula closely resembles the one obtained previously for internal division problems. The notable differences are as follows:

  • The two expressions are subtracted rather than added.
  • The 𝑚+𝑛 expression is replaced above by 𝑚−𝑛.

On the right-hand side above, the expression 𝑚+𝑛 for internal partitioning is replaced by 𝑚−𝑛 for external partitioning. Let’s derive the formula for the Cartesian coordinates of the partitioning point, given the coordinates 𝐴(𝑥,𝑦) and 𝐵(𝑥,𝑦): 𝑂𝑃=𝑚𝑚−𝑛(𝑥,𝑦)−𝑛𝑚−𝑛(𝑥,𝑦)=𝑚𝑚−𝑛𝑥,𝑚𝑚−𝑛𝑦−𝑛𝑚−𝑛𝑥,𝑛𝑚−𝑛𝑦=𝑚𝑚−𝑛𝑥−𝑛𝑚−𝑛𝑥,𝑚𝑚−𝑛𝑦−𝑛𝑚−𝑛𝑦=𝑚𝑥−𝑛𝑥𝑚−𝑛,𝑚𝑦−𝑛𝑦𝑚−𝑛.

This leads to 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

Fill in the blank: Given that 𝐶(−3,3) and 𝐷(4,−2), the 𝑥-axis divides 𝐶𝐷 in 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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