# 11.5: Distance and Midpoint Formulas

## 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 values (and the order of the values) are plugged in does not matter.

Point out that the midpoint of a segment is found by taking the average values of the and 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.*

and .

Hint: Plug the values of the two points into the distance formula; notice that parentheses were used around .