Find the Midpoint of a Line Segment

Finding the midpoint of a line segment is easy as long as you know the coordinates of the two endpoints. The most common way to do this is to use the midpoint formula, but there's another way to find the midpoint of a line segment if it's 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.

Steps

Use the Midpoint Formula

  1. Understand the midpoint. The midpoint of a line segment is the point that is located on the exact midpoint of the two endpoints. Therefore, it's the average of the two endpoints, which is the average of the two x-coordinates and the two y-coordinates.
  2. Learn the midpoint formula. The midpoint formula can be used by adding the x-coordinates of the two endpoints and dividing the result by two and then adding the y-coordinates of the endpoints and dividing them by two. This is how you will find the average of the x and y coordinates of the endpoints.[1] This is the formula: [(x1 + x2)/2,( y1 + y2)/2]
  3. Locate 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). Therefore, (x1, y1) = (5, 4) and (x2, y2) = (3, -4).
    • Note that either pair of coordinates can serve as (x1, y1) or (x2, y2) -- since you'll just be adding the coordinates and dividing by two, it doesn't matter which pair is first.
  4. Plug the corresponding 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]
  5. Solve. Once you've plugged the appropriate coordinates into the formula, all you have to do is the 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 endpoints (5,4) and (3, -4) is (4,0).

Find the Midpoint of Vertical or Horizontal Lines

  1. Find a vertical or horizontal line. Before you can use this method, you'll need to know how to locate a vertical or horizontal line. Here's how to spot it:[2]
    • A line is horizontal if the two y-coordinates of the endpoints are equal. For example, the line segment with the endpoints (-3, 4) and (5, 4) is horizontal.
    • A line is vertical if the two x-coordinates of the endpoints are equal. For example, the line segment with the endpoints (2, 0) and (2, 3) is vertical.
  2. Find the length of the segment. You can easily find the length of the segment just by counting how many horizontal spaces it takes up if it's horizontal, and counting how many vertical spaces it takes up if it's vertical. Here's how to do it:
    • The horizontal line segment with the end points (-3, 4) and (5, 4) is 8 units long. You can find this by counting the spaces it takes up or by adding the absolute values of the x-coordinates: |-3| + |5| = 8
    • The vertical line segment with the end points (2, 0) and (2, 3) is 3 units long. You can find this by counting the spaces it takes up or by adding the absolute values of the y-coordinates: |0| + |3| = 3
  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
  4. Count that value from either of the endpoints. This is the last step to finding the endpoint of the line segment. Here's how you do it:
    • To find the midpoint of the points (-3, 4) and (5, 4), just shift over 4 units either from the left or right to reach the middle of the segment. (-3, 4) shifted over 4 x-coordinates is (1, 4). You won't need to change the y-coordinates since you know 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), just shift over 1.5 units either from the top or bottom to reach the middle of the segment. (2, 0) shifted up 1.5 y-coordinates is (2, 1.5). You won't need to change the x-coordinates since you know the midpoint will be on the same x-coordinate as the endpoints. The midpoint of (2, 0) and (2, 3) is (2, 1.5).



Things You'll Need

  • Pencil
  • A sheet of paper
  • Ruler
  • Scissors

Related Articles

Sources and Citations

  1. ↑ http://www.mathplanet.com/education/algebra-1/radical-expressions/the-distance-and-midpoint-formulas
  2. ↑ http://www.regentsprep.org/Regents/math/geometry/GCG2/Lmidpoint.htm
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