Algebra

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# Midpoint Formula

## Definition of the Midpoint Formula

The midpoint formula gives us the center point on a line between two points. It is given as:**(x, y) = ( ^{(x1 + x2)}⁄_{2} , ^{(y1 + y2)}⁄_{2})**

The x and y coordinates used in the formula come from the two end points of the line being examined. This formula gives us the exact midpoint of a straight line.

### Where the Formula Comes From

**The midpoint formula takes the average of the line’s endpoints.** The first term in the formula sums x_{1} and x_{2}, then divides that sum by two. The second term in the formula sums y_{1} and y_{2}, then divides that sum by two.

We know that a straight line is simply a connection between two points in the coordinate system. Its midpoint is also considered a point in the coordinate system. This is why the midpoint formula is in the format of coordinates for a point. Take a look at the image above for a visual of how the midpoint formula is derived.

### Example Problems

**Problem 1:**

What is the midpoint of the line generated by the points (5 , 4) and (6 , 7)?

Solution:

(5 + 6)/2 = 5.5, and (4 + 7)/2 = 5.5**The midpoint of the line is (5.5, 5.5)**.

**Problem 2:**

A line has an endpoint at (0, 2) and its midpoint is at (8, 4). What is the other endpoint?

Solution:

Let’s work backwards through the midpoint formula. ^{0 + x2}/_{2} = 8, so x_{2} = 16. Then, ^{2 + y2}/_{2} = 4, so y_{2} = 6.**The other endpoint is (16, 6)**.

**Problem 3:**

A line has two endpoints given as (-3, -3) and (12, 5). What is the midpoint of the line?

Solution:

(-3 + 12)/2 = 4.5, and (-3 + 5)/2 = 1**The midpoint of the line is (4.5, 1)**.