A rotation on the coordinate plane turns a figure around a fixed point called the center of rotation. For many geometry problems, the center is the origin, (0, 0). Rotations help students understand symmetry, congruence, and transformations of shapes.
They also connect geometry to maps, graphics, engineering, and motion in the real world.
When a figure rotates about the origin, every point moves along a circular path centered at the origin. The distance from each point to the origin stays the same, but the coordinates change according to the angle and direction of rotation. The most common rotations are 90 degrees, 180 degrees, and 270 degrees counterclockwise.
These rotations have simple coordinate rules that make it possible to transform a whole polygon point by point.
Key Facts
- 90 degrees counterclockwise about the origin: (x, y) -> (-y, x)
- 180 degrees about the origin: (x, y) -> (-x, -y)
- 270 degrees counterclockwise about the origin: (x, y) -> (y, -x)
- 90 degrees clockwise about the origin is the same as 270 degrees counterclockwise: (x, y) -> (y, -x)
- A rotation preserves side lengths, angle measures, area, and shape, so the image is congruent to the original figure.
- For any rotation about the origin, the distance to the origin is unchanged: r = sqrt(x^2 + y^2)
Vocabulary
- Rotation
- A transformation that turns every point of a figure around a fixed center by a given angle.
- Center of rotation
- The fixed point around which a figure turns during a rotation.
- Image
- The new figure or point created after a transformation is applied.
- Preimage
- The original figure or point before a transformation is applied.
- Counterclockwise
- The direction opposite the movement of clock hands, usually treated as the positive direction for rotation angles.
Common Mistakes to Avoid
- Using the 90 degree rule backwards, such as changing (x, y) to (y, -x) for a 90 degree counterclockwise rotation. That rule is for 270 degrees counterclockwise or 90 degrees clockwise.
- Forgetting to change both signs in a 180 degree rotation. The correct rule is (x, y) -> (-x, -y), so both coordinates must become their opposites.
- Rotating the shape around the wrong center. The standard coordinate rules only work for rotations about the origin, not about another point unless the figure is shifted first.
- Assuming a rotation changes the size or shape of the figure. Rotations are rigid transformations, so lengths, angles, and area stay the same.
Practice Questions
- 1 Point A is at (4, 2). Find the coordinates of A after a 90 degrees counterclockwise rotation about the origin, then after a 180 degrees rotation about the origin.
- 2 Triangle ABC has A(1, 3), B(4, 2), and C(2, 6). Find the coordinates of A', B', and C' after a 270 degrees counterclockwise rotation about the origin.
- 3 A figure is rotated 180 degrees about the origin. Explain why the image is congruent to the original figure and why each point stays the same distance from the origin.