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Midpoint Calculator

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The midpoint calculator determines the point lying halfway between two points in the Cartesian coordinate system. This point is often useful in geometry. As a supplement to this calculator, we have written an article below discussing how to find the midpoint and what the midpoint formula is.

For a deeper understanding of how one coordinate changes relative to another, we recommend checking the average rate of change calculator.

What is a midpoint?

As the name suggests, a midpoint is a point that lies exactly halfway between two other points. For example, imagine we have a line segment that we want to divide into two equal parts. To do this, we must locate the center of this line segment, which is done by finding the midpoint. This midpoint is a point that is equidistant from both the endpoint coordinates.

To find out the midpoint, we can use a ruler. However, we can also use a formula that involves the coordinates of each endpoint of the line segment. The midpoint is essentially the average of the coordinates of the segment, resulting in a new coordinate point. Let's illustrate this concept below.

Midpoint formula

To calculate the midpoint between two coordinates, namely (x₁, y₁) and (x₂, y₂), we can use the midpoint formula. The midpoint coordinates, denoted as (x, y), are calculated by taking the average of the x-coordinates and the average of the y-coordinates like this:

x=x1+x22y=y1+y22\begin{align*} x& = \frac{x_1 + x_2}{2}\\[1em] y& = \frac{y_1 + y_2}{2} \end{align*}

In the following section, we will break down the process step by step.

How to find the midpoint

To find the midpoint between two coordinates, we should follow these steps:

  1. Identify the given coordinates: First, we need to identify the coordinates of the two points we want to find the midpoint between. Let's say we have the coordinates (x₁, y₁) and (x₂, y₂) as shown in the figure below.
Line segment showing midpoint coordinates.

  1. Apply the midpoint formula: Now, we will apply the midpoint formula to find the coordinates of the midpoint.

  2. For the x-coordinate of the midpoint x, we should add the x-coordinates of the two points and divide by 2:

x=x1+x22\qquad x = \frac{x_1 + x_2}{2}\\
  1. Similarly, for the y-coordinate of the midpoint y, we need to add the y-coordinates of the two points and divide by 2:
y=y1+y22\qquad y = \frac{y_1 + y_2}{2}

For example, imagine a line segment with coordinates (2, 4) and (6, 10). We can find the midpoint coordinates as follows:

x=2+62=82=4y=4+102=142=7\quad \begin{align*} x& = \frac{2 + 6}{2} = \frac{8}{2}=4\\[1.5em] y& = \frac{4 + 10}{2} = \frac{14}{2}=7\\ \end{align*}

How to use the midpoint calculator

Now, let's see how we can solve the same problem using the midpoint formula calculator.

  1. Enter the coordinates of the first point, e.g., (x₁ = 2, y₁ = 4).

  2. Input the coordinates of the second point, e.g., (x₂ = 6, y₂ = 10).

  3. The midpoint calculator will display the coordinates of the midpoint as (x = 4, y = 7).

  4. You can also calculate the coordinates of the endpoint if you input the coordinates of the midpoint and the other endpoint.

It is possible to divide a line segment into any given ratio, not just 1:1. Use our ratios of directed line segments calculator to learn how.

FAQs

How do I find the midpoint of a line segment?

To determine the midpoint of a line segment:

  1. Label the coordinates (x₁, y₁) and (x₂, y₂).
  2. Add together both x and y values separately.
  3. Divide each result by 2.
  4. The new values form the coordinates of the midpoint.

How do you find the endpoint with the midpoint and the other endpoint?

  1. Double your midpoint.
  2. Subtract your known endpoint to get the other. It doesn't matter if it's the upper or lower bound.
  3. Marvel at your mathematical skills!

What is the midpoint formula?

The midpoint formula calculates the coordinates of the midpoint of a line segment, given the coordinates of its endpoints. If we have coordinates (x₁, y₁) and (x₂, y₂), then the midpoint of these coordinates is determined by (x₁ + x₂)/2, (y₁ + y₂)/2.

How do I find the midpoint of a triangle?

To find the midpoint of a triangle, called its centroid, follow these steps:

  1. Find the midpoint of the sides of the triangle. If you know how to do this, skip to step 4.

  2. Measure the distance between the two endpoints, and divide the result by 2. This distance from either end is the midpoint of that line.

  3. Alternatively, add the two x coordinates of the endpoints and divide by 2. Do the same for the y coordinates. The results give you the coordinates of the midpoint.

  4. Draw a line between a midpoint and its opposite corner.

  5. Repeat for at least one other midpoint and corner pair, or both, for the highest degree of accuracy.

  6. Where all the lines meet is the centroid of the triangle.

What is the midpoint of a circle?

To find the midpoint, or center, of a circle, follow these instructions:

  1. Find two points on the circle that are opposite from each other, i.e., they are separated by the diameter of the circle.

  2. If you know their coordinates, add the two x-coordinates together and divide the result by 2. This is the x-coordinate of the center.

  3. Do the same for the two y-coordinates, which will give you the y-coordinate.

  4. Combine the two to get the center's coordinates.

  5. If you do not know the coordinates, measure the distance between the two points and half it.

  6. This half distance between one endpoint and the other is the midpoint.

How do I find the midpoint of a square?

To find the midpoint, or centroid, of a square, follow this simple guide:

  1. If you have the coordinates of two opposite corners of a square, add both x-coordinates together and divide the result by 2.

  2. Do the same for the y-coordinates.

  3. Use these two calculated numbers to find the center of the square, as they are its x- and y-coordinates, respectively.

  4. Alternatively, draw a line from one corner to the opposite corner, and another for the remaining pair.

  5. Where these two lines cross is the square’s centroid.

Do you round midpoints?

In general, you don’t round midpoints. You definitely do not for continuous data, as that point is a real point in a data set. For discrete data, you generally do not, instead noting that the midpoint is the value of both of the values on either side of the midpoint calculation.

How do you find the midpoint of a trapezoid?

You can find the midpoint, or centroid, of a trapezoid by one of two methods:

  1. Draw a line from one corner of the trapezoid to its opposing corner.
  2. Do the same for the remaining pair of corners.
  3. Where these two lines cross is the centroid.
  4. Balance your trapezoid perfectly on its centroid!

Alternatively:

  1. Take the coordinates of two opposite sides.
  2. Add the x coordinates of these points together and divide by 2. This is the midpoint’s x coordinate.
  3. Repeat for the two y-coordinates, giving the midpoint’s y-coordinate.

What is the midpoint of (0,2) and (2,8)?

  1. Add 0 and 2 together to get 2.
  2. Divide the result by 2, which results in 1. This is the x-coordinate of the midpoint.
  3. Add 2 and 8 together, which is 10.
  4. Divide 10 by 2, the result of which is 5; this is the y-coordinate of the midpoint.
  5. Put the two coordinates together; the midpoint of (0, 2) and (2, 8) is (1, 5).

What is the midpoint of 30 and 60?

45. To find the midpoint of any two numbers, find the average of those two numbers by adding them together and dividing by 2. In this case, 30 + 60 = 90 and 90 / 2 = 45.

A graph showing how to find the midpoint of a segment on the Cartesian plane.

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