# 11.5: Distance and Midpoint Formulas

**At Grade**Created by: CK-12

## Learning Objectives

At the end of this lesson, students will be able to:

- Find the distance between two points in the coordinate plane.
- Find the missing coordinate of a point given the distance from another known point.
- Find the midpoint of a line segment.
- Solve real-world problems using distance and midpoint formulas.

## Vocabulary

Terms introduced in this lesson:

- distance
- equidistant
- midpoint

## Teaching Strategies and Tips

Use Examples 1 and 2 to show how the Pythagorean Theorem is used to derive the distance formula.

- Teachers are encouraged to use a picture in the derivation.

Use Examples 3-5 and *Review Questions* 7, 8, and 15-20 as thinking problems.

- Draw pictures to help.
- Contrast these problems with the mechanical exercises of Example 2 and
*Review Questions*1-6.

Point out that because of the squares in the distance formula, the order in which the \begin{align*}x-\end{align*}values (and the order of the \begin{align*}y-\end{align*}values) are plugged in does not matter.

Point out that the midpoint of a segment is found by taking the average values of the \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}values of the endpoints.

In Example 9, suggest that students express their answers in radical form.

## Error Troubleshooting

General Tip: For points with negative coordinates, remind students about the minus sign in the distance formula.

Example:

*Find the distance between the two points.*

\begin{align*}(-2, 5)\end{align*} and \begin{align*}(3, -8)\end{align*}.

Hint: Plug the values of the two points into the distance formula; notice that parentheses were used around \begin{align*}-8\end{align*}.

\begin{align*}d = \sqrt{(-2-3)^2 + (5-(-8))^2}\end{align*}