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The Midpoint Formula

In One Dimension

On a number line , the number halfway between x 1 and x 2 is

x 1 + x 2 2

Example 1:

Find the midpoint between − 1 and 4 .

Use the formula. The midpoint is

− 1 + 4 2 = 3 2 = 1.5 .

Example 2:

If 0.5 is the midpoint of P R ¯ and the coordinate of P is − 4 , find the coordinate of R .

Use the formula.

− 4 + x 2 2 = 0.5

To begin solving, multiply both sides by 2 .

− 4 + x 2 = 1

Next, add 4 to both sides.

x 2 = 5

So, the coordinate of R is 5 .

In Two Dimensions

Suppose you are given two points in the plane ( x 1 , y 1 ) and ( x 2 , y 2 ), and asked to find the point halfway between them. The coordinates of this midpoint will be:

( x 1 + x 2 2 , y 1 + y 2 2 )

An easy way to think about this is that the x -coordinate of the midpoint is the average of the x -coordinates of the two points, and likewise with the y -coordinate.

Example 1:

Find the midpoint between ( − 2 , 5 ) and ( 7 , 7 ) .

Use the formula. The coordinates of the midpoint are:

( − 2 + 7 2 , 5 + 7 2 )

Simplify.

( 2.5 , 6 )

Example 2:

If Q ( 2 , − 2 ) is the midpoint of P R ¯ and P has the coordinates ( − 6 , − 6 ) , find the coordinates of R .

Use the formula to write and solve two equations for the coordinates of R .

Q ( 2 , − 2 ) = ( − 6 + x 2 2 , − 6 + y 2 2 )

First, find the x -coordinate.

2 = − 6 + x 2 2 4 = − 6 + x 2 10 = x 2

Then, find the y -coordinate.

− 2 = − 6 + y 2 2 − 4 = − 6 + y 2       2 = y 2

So, the coordinates of R are ( 10 , 2 ) .

In Three Dimensions

It's pretty easy to guess based on the formula for two dimensions!

In 3 -dimensional space, the midpoint between ( x 1 , y 1 , z 1 ) and ( x 2 , y 2 , z 1 ) is

( x 1 + x 2 2 , y 1 + y 2 2 , z 1 + z 2 2 )