Rotations of 180 Degrees

Examples, videos, and solutions to help Grade 8 students learn the characteristics of 180° rotations.

New York State Common Core Math Grade 8, Module 2, Lesson 6

Worksheets for Grade 8

Student Outcomes

Students learn that a rotation of 180 degrees moves a point on the coordinate plane (a, b), to (-a, -b).

Students learn that a rotation of 180 degrees around a point, not on the line, produces a line parallel to the given line.

Example 1

The picture below shows what happens when there is a rotation of 180° around center O.

Example 2

The picture below shows what happens when there is a rotation of 180 around center O the origin of the coordinate plane.

Exercises

1. Using your transparency, rotate the plane 180 degrees, about the origin. Let this rotation be R_O. What are the coordinates of R_O (2, -4) ?

2. Let R_O be the rotation of the plane by 180 degrees, about the origin. Without using your transparency, find R_O (-3, 5).

3. Let R_O be the rotation of 180 degrees around the origin. Let L be the line passing through (-6, 6) parallel to the x-axis. Find R_O (L). Use your transparency if needed.

4. Let R_O be the rotation of 180 degrees around the origin. Let L be the line passing through (7, 0) parallel to the y-axis. Find R_O (L). Use your transparency if needed.

5. Let R_O be the rotation of 180 degrees around the origin. Let L be the line passing through (0, 2) parallel to the x-axis. Is L parallel to R_O (L)?

6. Let R_O be the rotation of 180 degrees around the origin. Let L be the line passing through (4, 0) parallel to the y-axis. Is L parallel to R_O (L)?

7. Let R_O be the rotation of 180 degrees around the origin. Let L be the line passing through (0, -1) parallel to the x-axis. Is L parallel to R_O (L)?

8. Let R_O be the rotation of 180 degrees around the origin. Is L parallel to R_O (L)? Use your transparency if needed.

9. Let R_O be the rotation of degrees around the origin. Is L parallel to R_O(L)? Use your transparency if needed.

Proof the Theorem: Let O be a point not lying on a given line L. Then the 180-degree rotation around O maps L to a line parallel to L.

Lesson Summary

A rotation of 180 degrees around O is the rigid motion so that if P is any point in the plane P, O and Rotation (P) are collinear (i.e., lie on the same line).

Given a 180-degree rotation, R_O around the origin O of a coordinate system, and a point P with coordinates (a, b), it is generally said that R_O(P) is the point with coordinates (-a, -b).

Theorem. Let O be a point not lying on a given line L. Then the 180-degree rotation around O maps L to a line parallel to L.

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