Rotations
Overview
A rotation turns a figure about a fixed point. This lesson covers rotations in the coordinate plane: the direction conventions for positive and negative angle measures, the standard 90°, 180°, and 270° rotations about the origin, rotations about any point (done by translating the center to the origin, rotating, and translating back), and rotational symmetry — when a figure can be rotated by 180° or less and map directly onto itself.
Vocabulary
- Rotation
- A transformation in which a figure is turned about a fixed point. Rays drawn from the center of rotation to a point and its image form the angle of rotation.
- Center of Rotation
- The fixed point about which a figure is turned. It may be the origin, the center of the figure, or any other defined point.
- Angle of Rotation
- The angle formed by rays from the center of rotation to a point and to its image. A positive measure indicates a counterclockwise rotation; a negative measure indicates a clockwise rotation.
- Rotational Symmetry
- A figure has rotational symmetry if it can be rotated about its center by 180° or less and the resulting image maps directly onto the original.
Main Ideas
- Positive angle measures rotate a figure counterclockwise, and negative measures rotate it clockwise. A rotation of −90° is the same as a rotation of 270°.
- About the origin: a 90° counterclockwise rotation sends (x, y) → (−y, x); 180° sends (x, y) → (−x, −y); 270° counterclockwise sends (x, y) → (y, −x).
- To rotate about any other point: translate so the center lands at the origin, apply the rotation, then translate back by the opposite amount.
- A rotation is a rigid motion — the image is congruent to the preimage, and every point stays the same distance from the center of rotation.
- Familiar symmetry: a rectangle maps onto itself at 180°, a square at 90° and 180°, and an equilateral triangle at 120°.
Lesson Notes
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Lesson Video
Watch the lesson video.
Interactive Activity
Rotate figures about the origin and other points, and explore rotational symmetry.
Practice Problems
Want More Practice?
Download the practice worksheet for this lesson, work the problems, then check your work with the solutions.