The Midpoint Between Two Points

# The Midpoint Between Two Points

If $A$ has coordinates $(x_1, y_1)$ and $B$ has coordinates $(x_2, y_2)$, then the midpoint of $A$ and $B$ is said to be a point $C$ that cuts the distance between $A$ and $B$ into 2 equal parts.

For example, the following diagram illustrates the midpoint of two points:

Conceptually, the midpoint can be calculated by taking the averages of the $x$-coordinates and the averages of the $y$-coordinates, and thus, the following gives a formula for calculating the midpoint:

(1)
\begin{align} \mathrm{midpoint} = \left ( \frac{x_1 + x_2}{2} , \frac{y_1 + y_2}{2} \right ) \end{align}

For example, consider the points $(2, 3)$ and $(6, 9)$. Applying our formula, we get that the coordinates of the midpoint are:

(2)
\begin{align} \mathrm{midpoint} = \left ( \frac{2 + 6}{2} , \frac{3 + 9}{2} \right ) \\ \mathrm{midpoint} = \left ( \frac{8}{2} , \frac{12}{2} \right ) \\ \mathrm{midpoint} = (4, 6) \end{align}

## Example Questions

• 1. Calculate the midpoint of $(2, 1)$ and $(0, 9)$.
• 2. Calculate the midpoint of $(-5, 6)$ and $(2, -4)$.
• 3. Given the points $(0, 4)$ and $(0, 10)$, show that $(0, 7)$ are the coordinates of the midpoint without using the midpoint formula.