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The Midpoint Formula (page 1 of 2)


Sometimes you need to find the point that is exactly between two other points. For instance, you might need to find a line that bisects (divides into equal halves) a given line segment. This middle point is called the "midpoint". The concept doesn't come up often, but the Formula is quite simple and obvious, so you should easily be able to remember it for later.

Think about it this way: If you are given two numbers, you can find the number exactly between them by averaging them, by adding them together and dividing by two. For example, the number exactly halfway between 5 and 10 is  [5 + 10]/2  =  15/2   = 7.5.

The Midpoint Formula works exactly the same way. If you need to find the point that is exactly halfway between two given points, just average the x-values and the y-values.

  • Find the midpoint between (1, 2) and (3, 6).

    Apply the Midpoint Formula:

      midpoint is (1, -2)

    So the answer is P = (1, 2).

Technically, the Midpoint Formula is the following:

    [(x_1 + x_2)/2 , (y_1 + y_2)/2]

But as long as you remember that you're averaging the two points' x- and y-values, you'll do fine. It won't matter which point you pick to be the "first" point you plug in.

  • Find the midpoint between (6.4, 3) and (10.7, 4).

    Apply the Midpoint Formula:

      P = (-2.15, 3.5)

    So the answer is P = (2.15, 3.5)

  • Find the value of p so that (2, 2.5) is the midpoint between (p, 2) and (1, 3).

    I'll apply the Midpoint Formula:   Copyright Elizabeth Stapel 2000-2011 All Rights Reserved

      ( [p - 1]/2 , 2.5 ) = ( -2, 2.5 ]

    This reduces to needing to figure out what p is, in order to make the x-values work:

      p = -3

    So the answer is p = 3.

Let's do some more examples....

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Cite this article as:

Stapel, Elizabeth. "The Midpoint Formula." Purplemath. Available from
    http://www.purplemath.com/modules/midpoint.htm. Accessed
 

 



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