## The Proof of the Midpoint Formula

**Introduction**

The midpoint is the middle point of a segment. It is equidistant from both end points. The midpoint formula is used to determine the coordinates of the midpoint of a segment. This formula is familiar to middle school and high school students, however most books do not discuss its proof. In this post, we discuss the mathematical proof of the midpoint formula.

The slope and distance formulas are needed to prove the midpoint formula. Recall that the slope of the line containing the two points and with coordinates and respectively is

.

In addition, the distance between is

.

and its midpoint is

.

We use these formulas to prove the midpoint formula theorem.

**Theorem**

If segment has endpoints and , its midpoint is

.

**Proof**

Let be the coordinates of and be the coordinates of . In showing that with coordinates

is the midpoint of , we have to show that (1) and (2) is on (Why?).

For (1), we show that the distance of is equal to the distance of .

Distance of :

Distance of :

Therefore, .

For (2), we have to show that the slope of is equal to the slope of .

Slope of :

Slope of :

The slope of and are equal and , therefore is the midpoint of .

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