Geometry / Unit 4: Transformations
4.3 Rotations
Notes
A rotation turns a figure a set angle around a fixed point, the center of rotation. Counterclockwise is the positive direction.
- About the origin, $90^\circ$ counterclockwise: $(x, y) \to (-y, x)$.
- $180^\circ$: $(x, y) \to (-x, -y)$.
- $270^\circ$ counterclockwise (same as $90^\circ$ clockwise): $(x, y) \to (y, -x)$.
- Rotations are rigid motions and keep orientation.
- Example: rotating $(3, 1)$ by $90^\circ$ counterclockwise about the origin gives $(-1, 3)$.
Practice
Rotate $(4, 2)$ by $90^\circ$ counterclockwise about the origin.
$(x, y) \to (-y, x)$ gives $(-2, 4)$.
Rotate $(-3, 5)$ by $180^\circ$ about the origin.
A $180^\circ$ rotation negates both coordinates.
Rotate $(6, -1)$ by $90^\circ$ clockwise about the origin.
$90^\circ$ clockwise is $(x, y) \to (y, -x)$, giving $(-1, -6)$.
Ready for more? Take the Unit 4 test.