# Lesson Explainer: Midpoint on the Coordinate Plane Mathematics

In this explainer, we will learn how to find the coordinates of a midpoint between two points or those of an endpoint on the coordinate plane.

In geometry, and many other areas of math, we will often need to find the center of a line segment, that is, the point on the line segment equidistant from both endpoints. This is called the midpoint of the line segment. For example, the center of a circle is the midpoint of any of the diameters of the circle. Another example is when finding the medians of triangles given the coordinates of the vertices.

Before we determine how to find the midpoint of a line segment, we will formally define the midpoint.

### Definition: Midpoint of a Line Segment

The midpoint of a line segment is the point on the line segment equidistant (at equal distance) from and . In other words, and .

For example, let’s find the midpoint of the line segment between and . We sketch this line segment as follows.

The midpoint of this line segment is the point on it that is equidistant from both endpoints; in other words, it is halfway between both. Since this line is vertical, we can determine the length of the line segment as the difference in the -coordiantes; the line has length 4. Half of this value is 2, so the midpoint will lie 2 units away from both endpoints.

This is the point ; we can see it is a distance of 2 from both endpoints. Another way of saying this is that we took the mean of the -coordinates; we have , so it is halfway between these values.

We can use the same reasoning to find the midpoint of a horizontal line segment. For example, to find the midpoint of the line segment between and , we would take the mean of the -coordinates to find that the midpoint is .

We can then ask the question more generally, how would we find the midpoint of a line segment that is not vertical or horizontal? To do this, we will consider the line segment between and , which is neither horizontal nor vertical. We will call its midpoint , as shown.

We note that since is the midpoint of the line segment, both halves of the line segment have equal lengths. To determine the coordinates of , we construct the following right triangles using vertical and horizontal lines.

By noting that the horizontal lines are parallel and the vertical lines are parallel, we can use the line segment as a transversal cutting parallel lines to find the following corresponding angles.

These right triangles have the same angles and their hypotenuses have the same length, so by the ASA criterion they must be congruent. Since these triangles are congruent, their bases have the same length and their heights are also equal. We add in the following lines and points.

We see that is the midpoint of the horizontal line segment between and , so is the mean of the -coordinates, giving us

Similarly, is the midpoint of the vertical line segment between and , so is the mean of the -coordinates, giving us

We have shown that the midpoint of the line segment between and has the coordinates

It is worth noting that this formula works even if the line segment is vertical or horizontal. For example, applying the formula to our previous example of the line segment between and , we have , , , and , giving us a midpoint of

Note that since the -coordinates are equal, taking the mean does not change this value, so the formula will work for any line segment. We can summarize this result as follows.

### Theorem: Formula for the Midpoint of a Line Segment

The midpoint of the line segment between and has the coordinates

It is also worth point out that we often refer to the midpoint of the line segment between and as just the midpoint between and . Let’s now see some examples of applying this formula to determine the midpoint of two given points.

### Example 1: Finding the Halfway Point of Two Coordinates

On the graph, which point is halfway between and ?

### Answer

We recall that the midpoint of two points is the point on the line segment between them that is equidistant from both points. The midpoint of the line segment between and has the coordinates

In our case, we will set , , , and , giving us

We can add this point to the diagram.

We note that the point is three units up and two units left from the endpoint and it is also three units up and two units left from the point to the other endpoint. This confirms that the distance between and either endpoint is the same.

Hence, the midpoint has the coordinates .

### Example 2: Finding the Midpoint given the Endpoints

Given and , what are the coordinates of the midpoint of ?

### Answer

We recall that the midpoint of a line segment is the point on the line segment equidistant from the endpoints, so the midpoint of the line segment between and has the coordinates

To find the midpoint between and , we set , , , and :

Hence, the midpoint of has the coordinates .

In our next example, we will use the midpoint and an endpoint to determine the coordinates of the other endpoint.

### Example 3: Finding the Endpoint of a Line Segment

Given and , what are the coordinates of if is the midpoint of ?

### Answer

We recall that the midpoint of a line segment is the point on the line segment equidistant from both endpoints. There are two methods we can use to find the coordinates of . The first method uses the fact that both the horizontal and the vertical distance from to must be equal to the horizontal and the vertical distance from to .

The horizontal distance from to is given by the difference in their -coordinates:

This must be equal to the horizontal distance from to :

Similarly, the vertical distance from to is given by

This is equal to the vertical distance from to :

This gives us , and we can see this in the following diagram.

Alternatively, we can use the following formula for the midpoint between and :

We are given the coordinates of an endpoint of the line segment and the midpoint, and we need to determine the coordinates of the other endpoint. We will set , so and , then . We can substitute these values into the formula for the midpoint and set this equal to :

For the -coordinates to be equal, we have

Multiplying through by 2 gives

Adding 8 to both sides yields

Similarly, for the -coordinates to be equal,

Multiplying through by 2 gives

Adding 3 to both sides yields

Hence, has the coordinates .

In our next example, we will use the formula for a midpoint to determine unknow values in the coordinates of a midpoint.

### Example 4: Finding the Unknowns in the Coordinates of a Point Using the Midpoint Formula

Find the values of and so that is the midpoint of the line segment between and .

### Answer

We recall that the midpoint between and has the coordinates

If we set and , then we can substitute the coordinates of these points into the formula for the midpoint and equate it to to see

Equating the -coordinates, we have

Equating the -coordinates, we have

Hence, and .

In our next example, we will apply the midpoint formula to a real-world problem involving the distance between a fountain, a house, and a road.

### Example 5: Finding the Midpoint in a Real-World Problem

A rectangular garden is next to a house along a road. In the garden is an orange tree 7 m from the house and 3 m from the road. There is also an apple tree, 5 m from the house and 9 m from the road. A fountain is placed halfway between the trees. How far is the fountain from the house and the road?

### Answer

Let’s start by sketching the information given. First, we sketch the rectangular garden, road, and house.

We are told that there are two trees; an orange tree 7 m from the house and 3 m from the road and an apple tree 5 m from the house and 9 m from the road, with a fountain placed half way between the trees as shown below.

To determine the distance the fountain is from the house and the road, we will write any point in the garden as a pair of coordinates of the form (distance from house, distance from road). For example, the point will be the point in the garden that touches the house and the road. So, the apple tree has the coordinates and the orange tree has the coordinates . Since the fountain is halfway between these points, it is their midpoint, and we recall that the midpoint between and has the coordinates

Substituting , , , and into the formula for the midpoint gives us

Hence, the fountain is 6 m from the house and 6 m from the road.

### Example 6: Finding an Endpoint given the Midpoint and the Other Endpoint

The origin is the midpoint of the straight segment . Find the coordinates of point if the coordinates of point are .

### Answer

We recall that the midpoint of a line segment is the point on the line segment equidistant from both endpoints. There are two methods we can use to find the coordinates of . The first method will use the fact that both the horizontal and the vertical distance from to must be equal to the horizontal and the vertical distance from to .

We see that the horizontal distance from to is given by the -coordinate of ,

Similarly, the vertical distance is given by the -coordinate of and is 4 units. Since is 6 units left of the midpoint and 4 units above , we must have that is 6 units right of and 4 units below , at the coordinates . We can see this in the following diagram.

Alternatively, we can find the coordinates of from the following formula for the midpoint of the line segment between and :

We are given the coordinates of an endpoint of the line segment and the midpoint, and we need to determine the coordinates of the other endpoint. We will set , so and , then . We can substitute these values into the formula for the midpoint and set this equal to the origin, :

For the -coordinates to be equal, we have

Multiplying through by 2 gives

Adding 6 to both sides yields

Similarly, for the -coordinates to be equal,

Multiplying through by 2 gives

Subtracting 4 from both sides yields

Hence, has the coordinates .

Let’s finish by recapping some of the important points of this explainer.

### Key Points

• The midpoint of a line segment is the point in the line segment equidistant from and . In other words, and . This is also referred to as the midpoint between and .
• The midpoint of the line segment between and has the coordinates

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