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.
Terms introduced in this lesson:
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 x−values (and the order of the y−values) are plugged in does not matter.
Point out that the midpoint of a segment is found by taking the average values of the x− and y−values of the endpoints.
In Example 9, suggest that students express their answers in radical form.
General Tip: For points with negative coordinates, remind students about the minus sign in the distance formula.
Find the distance between the two points.
(−2,5) and (3,−8).
Hint: Plug the values of the two points into the distance formula; notice that parentheses were used around −8.