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, and , using the notation . contains all the points on the straight line between and .

To help us understand this definition, we can consider a line segment drawn on the coordinate plane with endpoints and .

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 by halving each of the horizontal and vertical distances between and .

### Recap: The Midpoint of a Line Segment

We can find the midpoint, , of a line segment between and using

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 and are and respectively. Determine the coordinates of the points that divide into four equal parts.

### Answer

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

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

We recall that the midpoint, , of a line segment between coordinates and is given by

To find the midpoint, , of , we substitute the coordinates of for the values and the coordinates of for the values, giving Thus, the coordinates of are .

Next, we find the midpoint, , of . Substituting the coordinates and for the , and values, respectively, gives

Finally, we find the midpoint, , of . Using the coordinates and gives

Thus, we have found the coordinates of , , and , which divide into 4 equal parts, as

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 and , then divides by the ratio .

### Answer

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

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

The movement from point to point is the vector .

The magnitudes of vectors and are their lengths. We are given that ; therefore, we can write that

We can divide into 3 equal pieces.

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

Thus, it would not be true that . Therefore, must be at the point that is closer to than .

In this way, and

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

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

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 and , vector tells us the relative direction of point with respect to point , as well as the distance between the two points. In particular, does not have to begin at or end at , 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 partitions in the ratio , this means that point lies on the line segment and the ratio of the magnitudes of the vectors satisfies

In other words, if is length units, would be equal to length units, which leads to

In general, it would be difficult to use only the equation above to find the coordinates of the partitioning point . However, this equation does not contain the information that lies on the line segment . In particular, this means that has the same direction as . 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 . This leads to

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 and each as a difference of two position vectors:

Substituting these expressions into the original formula leads to

Rearranging the equation so that is the subject, we obtain

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

Let be a point on line segment , partitioning it in the ratio . Then, the position vector is given by

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 and . Then, we can write the corresponding position vectors

Substituting these expressions into the formula above, we obtain

We arrive at the following formula.

### Theorem: The Section Formula

If we have distinct points and and the point divides such that , then has the coordinates

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 and are and , respectively, find the coordinates of point that divides internally by the ratio .

### Answer

We can sketch this directed line segment as shown.

We can apply the following formula to find point that divides internally in the ratio . This means that and the ratio will be given as .

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

If and and point divides such that , then has the coordinates

For our problem, has coordinates and has coordinates . We can substitute these coordinates into the formula for and respectively.

The ratio values can be substituted for and respectively.

Therefore, we will have the coordinates of as

Simplifying, we have

Thus, the coordinates of point that divides internally by the ratio are .

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 divides externally in the ratio , where . 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 , with magnitude length units, is the larger vector compared to , with magnitude length units. By subtraction, we can see that the length of is equal to units. This leads to

As in the previous context, and have the same direction, so we can write

As we have done for the internal partitioning problem, we can calculate the formula for the position vector of :

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 expression is replaced above by .

On the right-hand side above, the expression for internal partitioning is replaced by for external partitioning. Let’s derive the formula for the Cartesian coordinates of the partitioning point, given the coordinates and :

This leads to the following formula.

### Theorem: The Section Formula with External Division

If we have distinct points and and point divides such that , then has the coordinates

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 and , find in vector form the coordinates of point that divides externally in the ratio .

### Answer

We can begin by sketching the points and and extending the directed line segment to point that divides externally. We can write that .

We recall the section formula for external division.

If we have distinct points and and point divides such that , then has the coordinates

We can substitute the values and for the and values, respectively, and the ratio values for and into the section formula to find the coordinates of . This gives us

As we are asked to give our answer in vector form, we can give the position vector of as

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 and , the -axis divides in the ratio .

### Answer

We can begin by plotting the coordinates and and sketching the vector .

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

The slope, , of a line joining two points and can be found using

Therefore, the slope between and is given by

We can then use the point–slope form of a line such that, given a point and the slope, , we can write the equation of the line as

We can substitute the coordinates of either or into this form, so using for the values and , we have

We can then multiply both sides by 7 and expand the parentheses on the right-hand side, giving

Rearranging to write this in the general form of the equation of a line, , we have

We recall that a line crosses the -axis when , so substituting this into the equation and simplifying gives

We have now calculated that the line segment crosses the -axis at the point .

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

To do this, we can use the section formula for internal division of a line segment. If we have distinct points and and point divides such that , then has the coordinates

In this question, we know points and and point that divides . We need to calculate the ratio values of and .

Substituting and for the and values, respectively, into the section formula, we have

We know that has coordinates , so we can write

Evaluating the -coordinates, we have

We can cross multiply and simplify to write an expression for in terms of as

The ratio of , so .

Thus, we can give the answer that the -axis divides in the ratio

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

We recall that the distance formula for finding the distance, , between two points and is given by

To find the length of , , we substitute the values of and for the and values to give

To find the length of , , we substitute and for the and values, which gives

We can then write the ratio of as

Multiplying both sides of the ratio by 5, and then dividing by , gives

Thus, we have confirmed our answer, that the -axis divides by the ratio .

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 to city . Its first stop is at , which is halfway between the cities. Its second stop is at , which is two-thirds of the way from to . What are the coordinates of and ?

### Answer

We are given that city has coordinates and city has coordinates . Firstly, we need to find the coordinates of city , halfway between these.

We can make use of the formula for the midpoint of a line. To find the midpoint, , of a line segment between two points and we can use

Substituting and into this formula gives the midpoint, , as

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

We can use the formula to partition a line segment in a given ratio. If and and point divides such that , then has the coordinates

In this question, we have , , and point that divides in the ratio . Substituting these into the formula to find gives

As a useful check of our answer, we can consider the lengths of and by applying the distance formula. To find the distance, , between two points and , we calculate

To calculate the length of , we can substitute the coordinates and into the formula to give

To calculate the length of , we substitute and into the distance formula, giving

Thus, we can write the ratio of lengths as

We were given that is two-thirds of the way from to . Therefore, we have confirmed that has the coordinates .

We can give the answer that the coordinates of and are

### Key Points

- The midpoint, , of the line segment between and is given by
- If we have distinct points and and point divides internally such that , then has the coordinates
- When answering problems involving the partitioning of a line segment, we need to be careful to establish the correct order of the ratio. If is split by point in the ratio , then the ratio of will be . If is split by point in the ratio , then the ratio of will instead be .
- If we have distinct points and and point divides externally such that , then has the coordinates