How to find the midpoint

Here you will learn about how to find the midpoint, including finding the midpoint of a line segment using Cartesian coordinates, and finding a missing endpoint when the midpoint and other endpoint is given.

Students will first learn about how to find the midpoint as a part of geometry in 8 th grade.

What is the midpoint of a line?

The midpoint of a line segment is a point that lies exactly halfway between two points. It is the same distance from each endpoint of the straight line segment.

We can find this one of two ways:

1. By inspection – this is easier with positive integer numbers
2. Using the midpoint of a line formula. Add the two x coordinates and divide by 2 to find the x coordinate of the midpoint, and add the two y coordinates and divide by 2 to find the y coordinate of the midpoint.

For example, if we graph the two given points (2,2) and (8,6) on a coordinate plane, the midpoint is exactly halfway between the two, and lies at (5, 4).

It is easy to see that 5 is halfway between 2 and 8, and 4 is halfway between 2 and 6. You can easily visualize this using the graph below.

Midpoint of a line formula

If it is not easy to spot the midpoint, or the coordinates involve fractions or negative numbers, we can use the midpoint formula.

If the points \mathrm{A}\left(x_{1}, y_{1}\right) and \mathrm{B}\left(x_{2}, y_{2}\right) are the endpoints of a line segment, then the midpoint of the line segment joining the points A and B is \left(\cfrac{x_{1}+x_{2}}{2}, \cfrac{y_{1}+y_{2}}{2}\right) .

This looks complicated when written algebraically, but you’re basically calculating the (mean) average of both the x values and the y values.

Add the two x coordinates and divide by 2 to find the x coordinate of the midpoint, and add the two y coordinates and divide by 2 to find the y coordinate of the midpoint.

For example, given two points A (-1,2) and B (2,4), the midpoint (M) is exactly halfway between the two, and lies at (0.5, 3).

To calculate the midpoint,

The mean average of the x coordinates is \cfrac{-1+2}{2}=\cfrac{1}{2}=0.5

The mean average of the y coordinates is \cfrac{2+4}{2}=\cfrac{6}{2}=3

We can also apply the Pythagorean Theorem to find the distance between two given points. To do this, we form a right-angled triangle with the line segment as the hypotenuse.

Pythagoras’ theorem tells us that h^{2}=2^{2}+3^{2}, and therefore the length of the hypotenuse is \sqrt{2^{2}+3^{2}}=\sqrt{13}.

If you study coordinate geometry further in high school, you may come across the general distance formula:

d=\sqrt{\left(x_{2}-x_{1}\right)+\left(y_{2}-y_{1}\right)} where (x_{1},y_{1}) and (x_{2},y_{2}) are the coordinates of the two points and d is the distance between them.

Finding a missing endpoint

You may not always be given both endpoints of the line. Sometimes you may be given one endpoint and the midpoint, and have to work out the other endpoint.

To get from the first endpoint (1,3) to the midpoint (3,7), move 2 in the x- direction and 4 in the y- direction. Repeat this again from the midpoint to find the coordinate of the other endpoint, which in this case would be (5,11).

What is the midpoint of a line?

Common Core State Standards

How does this relate to 8 th grade math?

• Grade 8: Geometry (8.G.B.8)
  Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

How to find the midpoint of a line segment

In order to find the midpoint of the line segment joining the endpoints \text{A} and \text{B:}

1. Find the average of the \textbf{x} coordinates of the two endpoints.
2. Find the average of the \textbf{y} coordinates of the two endpoints.
3. Write down the coordinates of the point.

How to find the midpoint examples

Example 1: two positive integer endpoints

Find the midpoint of the line segment joining the points (0,6) and (4, 10).

1. Find the average of the \textbf{x} coordinates of the two endpoints.

2Find the average of the \textbf{y} coordinates of the two endpoints.

3Write down the coordinates of the point.

In this case, it is quite easy to see the midpoint by inspection, particularly if working on a graph.

Example 2: two positive integer endpoints with a fractional answer

Find the midpoint of the line segment joining the points (1,5) and (6, 0).

Example 3: coordinate pairs containing negative numbers

Find the midpoint of the line segment joining the points (-2,7) and (4, 10).

Write down the coordinates of the point.

(1, 8.5)

Graphically,

Example 4: coordinates containing decimals

Find the midpoint of the line segment joining the points (0.5, 3) and (4, 2.5).

Write down the coordinates of the point.

(2.25, 2.75)

Graphically,

How to find a missing endpoint

In order to find a missing endpoint when given one endpoint and the midpoint:

1. Work out how to get from the given endpoint to the midpoint.
2. Repeat this to get from the midpoint to the missing endpoint.
3. Write down the coordinates of the missing endpoint.

Example 5: finding a missing endpoint when given one endpoint and the midpoint

A line segment joins the points A and B, and has midpoint M.

Point A has coordinates (4, 8) and point M has coordinates (6, 9).

Find the coordinates of point B.

Work out how to get from the given endpoint to the midpoint.

To get from A to M, we add 2 to the x- coordinate of A and add 1 to the y- coordinate of A.

Repeat this to get from the midpoint to the missing endpoint.

To get from M to B, we add 2 to the x- coordinate of M and add 1 to the y- coordinate of M.

Example 6: finding a missing endpoint with negative coordinates

A line segment joins the points A and B, and has midpoint M.

Point A has coordinates (-9, 4) and point M has coordinates (-6, -1).

Find the coordinates of point B.

Work out how to get from the given endpoint to the midpoint.

To get from A to M, we add 3 to the x- coordinate of A and subtract 5 from the y- coordinate of A.

Repeat this to get from the midpoint to the missing endpoint.

To get from M to B, we add 3 to the x- coordinate of M and subtract 5 from the y- coordinate of M.

Teaching tips for how to find the midpoint

• Use visual aids to help illustrate the concept, such as number lines. This can provide foundational understanding for the concept of midpoints, before using a coordinate plane.

• Provide students with step-by-step instructions on how to find the midpoint to refer back to when needed. These can be placed in a math journal or written on an anchor chart within the classroom.

• Teach students to use interactive technology, like a midpoint calculator, to assist students with finding the midpoint using different avenues.

Easy mistakes to make

• Finding the average of each point rather than the average of the \textbf{x} coordinates and average of the \textbf{y} coordinates
  For example, for the points (2, 3) and (5, 7), make sure you don’t do \cfrac{2+3}{2} and \cfrac{5+7}{2}.

• Using the midpoint formula when given one endpoint and the midpoint
  If one endpoint is (3, 4) and the midpoint is (6, 2), make sure you work out how you get from the endpoint to the midpoint and repeat this, rather than using the midpoint formula.

• Errors with negative number calculations
  If you’re not sure about your answer, draw a diagram and count the steps.

Related graphing linear equations lessons

• Graphing linear equations
• y = mx +b
• How to find the y- intercept
• How to find the slope of a line
• Distance formula
• Linear interpolation

The next lessons are

• Rate of change
• Systems of equations
• Number patterns
• Angles

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