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 𝐴𝐡 is the point 𝐢 on the line segment equidistant (at equal distance) from 𝐴 and 𝐡. In other words, 𝐢∈𝐴𝐡 and 𝐴𝐢=𝐡𝐢.

For example, let’s find the midpoint of the line segment between (1,1) and (1,5). 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 𝑦-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 (1,3); 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 𝑦-coordinates; we have 3=5+12, 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 (π‘₯,π‘Ž) and (π‘₯,π‘Ž), we would take the mean of the π‘₯-coordinates to find that the midpoint is ο€Όπ‘₯+π‘₯2,π‘ŽοˆοŠ§οŠ¨.

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 (π‘₯,𝑦) and (π‘₯,𝑦), which is neither horizontal nor vertical. We will call its midpoint 𝐢(𝑐,𝑐), as shown.

We note that since 𝐢 is the midpoint of the line segment, both halves of the line segment have equal lengths. To determine the coordinates of 𝐢, 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 (𝑐,𝑦) is the midpoint of the horizontal line segment between (π‘₯,𝑦) and (π‘₯,𝑦), so π‘οŠ§ is the mean of the π‘₯-coordinates, giving us 𝑐=π‘₯+π‘₯2.

Similarly, (π‘₯,𝑐) is the midpoint of the vertical line segment between (π‘₯,𝑦) and (π‘₯,𝑦), so π‘οŠ¨ is the mean of the 𝑦-coordinates, giving us 𝑐=𝑦+𝑦2.

We have shown that the midpoint of the line segment between (π‘₯,𝑦) and (π‘₯,𝑦) has the coordinates ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

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 (1,1) and (1,5), we have π‘₯=1, 𝑦=1, π‘₯=1, and 𝑦=5, giving us a midpoint of ο€Ό1+12,1+52=ο€Ό22,62=(1,3).

Note that since the π‘₯-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 (π‘₯,𝑦) and (π‘₯,𝑦) has the coordinates ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

It is also worth point out that we often refer to the midpoint of the line segment between 𝐴 and 𝐡 as just the midpoint between 𝐴 and 𝐡. 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 (1,8) and (5,2)?

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 (π‘₯,𝑦) and (π‘₯,𝑦) has the coordinates ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

In our case, we will set π‘₯=1, 𝑦=8, π‘₯=5, and 𝑦=2, giving us ο€Ό1+52,8+22=ο€Ό62,102=(3,5).

We can add this point to the diagram.

We note that the point (3,5) is three units up and two units left from the endpoint (5,2) and it is also three units up and two units left from the point (3,5) to the other endpoint. This confirms that the distance between (3,5) and either endpoint is the same.

Hence, the midpoint has the coordinates (3,5).

Example 2: Finding the Midpoint given the Endpoints

Given 𝐴(4,8) and 𝐡(6,6), what are the coordinates of the midpoint of 𝐴𝐡?

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 (π‘₯,𝑦) and (π‘₯,𝑦) has the coordinates ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

To find the midpoint between 𝐴(4,8) and 𝐡(6,6), we set π‘₯=4, 𝑦=8, π‘₯=6, and 𝑦=6: ο€Ό4+62,8+62=ο€Ό102,142=(5,7).

Hence, the midpoint of 𝐴𝐡 has the coordinates (5,7).

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 𝐴(βˆ’8,βˆ’3) and 𝐢(4,1), what are the coordinates of 𝐡 if 𝐢 is the midpoint of 𝐴𝐡?

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 𝐡. The first method uses the fact that both the horizontal and the vertical distance from 𝐴 to 𝐢 must be equal to the horizontal and the vertical distance from 𝐢 to 𝐡.

The horizontal distance from 𝐴 to 𝐢 is given by the difference in their π‘₯-coordinates: π‘₯βˆ’π‘₯=4βˆ’(βˆ’8)=12.

This must be equal to the horizontal distance from 𝐢 to 𝐡: 12=π‘₯βˆ’π‘₯12=π‘₯βˆ’416=π‘₯.

Similarly, the vertical distance from 𝐴 to 𝐢 is given by π‘¦βˆ’π‘¦=1βˆ’(βˆ’3)=4.

This is equal to the vertical distance from 𝐢 to 𝐡: 4=π‘¦βˆ’π‘¦4=π‘¦βˆ’15=𝑦.

This gives us 𝐡(π‘₯,𝑦)=𝐡(16,5), and we can see this in the following diagram.

Alternatively, we can use the following formula for the midpoint between (π‘₯,𝑦) and (π‘₯,𝑦): ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

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 𝐴(π‘₯,𝑦), so π‘₯=βˆ’8 and 𝑦=βˆ’3, then 𝐡(π‘₯,𝑦). We can substitute these values into the formula for the midpoint and set this equal to 𝐢(4,1): (4,1)=ο€Όβˆ’8+π‘₯2,βˆ’3+𝑦2.

For the π‘₯-coordinates to be equal, we have 4=βˆ’8+π‘₯2.

Multiplying through by 2 gives 8=βˆ’8+π‘₯.

Adding 8 to both sides yields π‘₯=16.

Similarly, for the 𝑦-coordinates to be equal, 1=βˆ’3+𝑦2.

Multiplying through by 2 gives 2=βˆ’3+𝑦.

Adding 3 to both sides yields 𝑦=5.

Hence, 𝐡 has the coordinates (16,5).

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 π‘Ž and 𝑏 so that (βˆ’2π‘Ž,2π‘Ž+𝑏) is the midpoint of the line segment between (βˆ’2,βˆ’3) and (2,11).

Answer

We recall that the midpoint between 𝐴(π‘₯,𝑦) and 𝐡(π‘₯,𝑦) has the coordinates ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

If we set 𝐴(βˆ’2,βˆ’3) and 𝐡(2,11), then we can substitute the coordinates of these points into the formula for the midpoint and equate it to (βˆ’2π‘Ž,2π‘Ž+𝑏) to see (βˆ’2π‘Ž,2π‘Ž+𝑏)=ο€Όβˆ’2+22,βˆ’3+112(βˆ’2π‘Ž,2π‘Ž+𝑏)=ο€Ό02,82(βˆ’2π‘Ž,2π‘Ž+𝑏)=(0,4).

Equating the π‘₯-coordinates, we have βˆ’2π‘Ž=0π‘Ž=0.

Equating the 𝑦-coordinates, we have 2π‘Ž+𝑏=42(0)+𝑏=4𝑏=4.

Hence, π‘Ž=0 and 𝑏=4.

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 𝑂(0,0) will be the point in the garden that touches the house and the road. So, the apple tree has the coordinates (5,9) and the orange tree has the coordinates (7,3). Since the fountain is halfway between these points, it is their midpoint, and we recall that the midpoint between 𝐴(π‘₯,𝑦) and 𝐡(π‘₯,𝑦) has the coordinates ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

Substituting π‘₯=5, 𝑦=9, π‘₯=7, and 𝑦=3 into the formula for the midpoint gives us fountain=ο€Ό5+72,9+32=(6,6).

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 𝐴𝐡. Find the coordinates of point 𝐡 if the coordinates of point 𝐴 are (βˆ’6,4).

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 𝐡. The first method will use the fact that both the horizontal and the vertical distance from 𝐴 to 𝑂 must be equal to the horizontal and the vertical distance from 𝑂 to 𝐡.

We see that the horizontal distance from 𝐴 to 𝑂 is given by the π‘₯-coordinate of 𝐴, |βˆ’6|=6.

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

Alternatively, we can find the coordinates of 𝐡 from the following formula for the midpoint of the line segment between (π‘₯,𝑦) and (π‘₯,𝑦): ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

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 𝐴(π‘₯,𝑦), so π‘₯=βˆ’6 and 𝑦=4, then 𝐡(π‘₯,𝑦). We can substitute these values into the formula for the midpoint and set this equal to the origin, (0,0): (0,0)=ο€Όβˆ’6+π‘₯2,4+𝑦2.

For the π‘₯-coordinates to be equal, we have 0=βˆ’6+π‘₯2.

Multiplying through by 2 gives 0=βˆ’6+π‘₯.

Adding 6 to both sides yields π‘₯=6.

Similarly, for the 𝑦-coordinates to be equal, 0=4+𝑦2.

Multiplying through by 2 gives 0=4+𝑦.

Subtracting 4 from both sides yields 𝑦=βˆ’4.

Hence, 𝐡 has the coordinates (6,βˆ’4).

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

Key Points

  • The midpoint of a line segment 𝐴𝐡 is the point 𝐢 in the line segment equidistant from 𝐴 and 𝐡. In other words, 𝐢∈𝐴𝐡 and 𝐴𝐢=𝐡𝐢. This is also referred to as the midpoint between 𝐴 and 𝐡.
  • The midpoint of the line segment between (π‘₯,𝑦) and (π‘₯,𝑦) has the coordinates ο€Όπ‘₯+π‘₯2,𝑦+𝑦2.

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