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.

Solutions

See solutions

• 1. Applying our formula we get that:

(3)

\begin{align} \mathrm{midpoint} = \left ( \frac{2 + 0}{2}, \frac{1 + 9}{2} \right ) \\ \mathrm{midpoint} = (1, 5) \end{align}

• 2. Applying our formula we get that:

(4)

\begin{align} \mathrm{midpoint} = \left ( \frac{-5 + 2}{2}, \frac{6 - 4}{2} \right ) \\ \mathrm{midpoint} = \left ( - \frac{3}{2} , 1 \right ) \end{align}

• 3. We note that all three of these points lie on the line $y = 0$. We note the distance between $(0, 4)$ and $(0, 10)$ is 6. Furthermore, we know that the distance between $(0,4)$ and $(0, 7)$ is 3, and the distance between $(0, 7)$ and $(0, 10)$ is 3. Since $(0, 7)$ splits the distance between the first two points in half equally and lines on the line segment of the distance between them, $(0, 7)$ must be the midpoint.