In this lesson, the midpoint between two points whose coordinates
are known will be found. We will also develop a general formula for determining the coordinates
of the midpoint and go through several examples.
Suppose it is desired to find the midpoint between the points (1, 2) and (3, -2) shown on the grid below. The Midpoint Formula:
To do this, we first look at a number line and find the midpoint between x = 1 and x = 3.
The principle that we apply will give us a general formula for the midpoint between any two points with given coordinates. The point that is exactly halfway between 1 and 3 on this one-dimensional number line is 2. This can be found by averaging the 2 coordinates:
If we apply the averaging strategy to our two points, we have: x = .
Therefore, the midpoint between (1, 2) and (3, –2) is (2, 0).
We can generalize the method used above. The midpoint between any two points is given by . This is known as “the midpoint formula.” Let's practice:
- What is the midpoint between the points (5, 6) and (– 12, 40)?
We apply the midpoint formula:
- If the midpoint between (1, 4) and (x, 10) is (–4, 7), what is the value of x?
We apply the midpoint formula for the 1st coordinate: which gives us x = – 9.