How to Find the Midpoint of a Line Segment: 9 Steps (with Pictures)

2024 Author: Jason Gerald | [email protected]. Last modified: 2024-01-15 08:07

Finding the midpoint of a line segment is easy as long as you know the coordinates of the two endpoints of the line. The most common way to find it is to use the midpoint formula, but there are other ways to find the midpoint of a line segment if the line is vertical or horizontal. If you want to know how to find the midpoint of a line segment in just a few minutes, just follow these steps.

Step

Method 1 of 2: Using the Midpoint Formula

Step 1. Understand about the midpoint

The midpoint of a line segment is the point that lies exactly in the middle of the two endpoints. Thus, the midpoint is the average of the two endpoints, which is the average of the two x-coordinates and the two y-coordinates.

Step 2. Learn the midpoint formula

The midpoint formula can be used by adding up the x-coordinates of the two endpoints and dividing the result by two, and then adding up the y-coordinates of the endpoints and dividing by two. This is how you find the average of the x and y coordinates of the endpoints. Here's the formula: [(x₁ +x₂)/2, (y₁ + y₂)/2]

Step 3. Find the coordinates of the endpoints

You can't use the midpoint formula without knowing the x and y coordinates of the endpoints. In this example, you want to find the midpoint, point O, which is between the two endpoints M (5, 4) and N (3, -4). Thus, (x₁, y₁) = (5, 4) and (x₂, y₂) = (3, -4).

Note that any pair of coordinates can be (x₁, y₁) or (x₂, y₂) -- since you're just adding the coordinates and dividing by two, it doesn't matter which pair of coordinates comes first.

Step 4. Plug the respective coordinates into the formula

Now that you know the coordinates of the endpoints, you can plug them into the formula. Here's how you do it:

[(5 + 3)/2, (4 + -4)/2]

Step 5. Finish

Once you've plugged the exact coordinates into the formula, all you have to do is do some simple arithmetic that will give you the midpoint of the two line segments. Here's how you do it:

• [(5 + 3)/2, (4 + -4)/2] =
• [(8/2), (0/2)] =
• (4, 0)
• The midpoint of the ends of the points (5, 4) and (3, -4) is (4, 0).

Method 2 of 2: Finding the Midpoint of Vertical and Horizontal Lines

Step 1. Look for vertical or horizontal lines

Before you can use this method, you need to know how to define vertical or horizontal lines. Here's how to find out:

• A line is considered horizontal if the two y-coordinates of its endpoints are the same. For example, a line segment with endpoints (-3, 4) and (5, 4) is horizontal.

  

• A line is considered vertical if the two x-coordinates of its endpoints are the same. For example, a line segment with endpoints (2, 0) and (2, 3) is vertical.

  

Step 2. Find the length of the segment

You can easily find the length of the segment simply by calculating the number of horizontal distances from the ends of the point if the line is horizontal, and counting the number of vertical distances from the ends of the point if the line is vertical. Here's how to do it:

• The horizontal line segment with the endpoints (-3, 4) and (5, 4) has a length of 8 units. You can find it by calculating the distance or by adding the absolute values of the x coordinates: |-3| + |5| = 8

  

• A vertical line segment with endpoints (2, 0) and (2, 3) has a length of 3 units. You can find it by calculating the distance or adding the absolute value of the y-coordinate: |0| + |3| = 3

  

Step 3. Divide the length of the segment by two

Now that you know the length of the line segment, you can divide it by two.

• 8/2 = 4

  

• 3/2 = 1, 5

  

Step 4. Calculate the value from any endpoint

This step is the last step to find the end point of the line segment. Here's how you do it:

• To find the midpoint of the points (-3, 4) and (5, 4), simply move 4 units from either the left or right to reach the midpoint of the line segment. (-3, 4) is shifted by 4 units of its x-coordinate to (1, 4). You don't need to change the y-coordinate because you know that the midpoint will be on the same y-coordinate as the endpoints. The midpoint of (-3, 4) and (5, 4) is (1, 4).

  

• To find the midpoint of the points (2, 0) and (2, 3), simply move 1.5 units from both the top and bottom to reach the midpoint of the line segment. (2, 0) is shifted by 1,5 its y-coordinate units to (2, 1, 5). You don't need to change the x-coordinates because you know that the midpoints will be at the same x-coordinates as the endpoints. The midpoint of (2, 0) and (2, 3) is (2, 1, 5).