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 x1 and x2, then divides that sum by two. The second term in the formula sums y1 and y2, 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 x2 = 16. Then, 2 + y2/2 = 4, so y2 = 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).

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