Integrated Math 1 / Unit 8: Transformations and Congruence
8.4 Rotations
Notes
A rotation turns a figure about a fixed point (the center) by a given angle and direction.
- About the origin, counterclockwise: $90^\circ$: $(x, y) \to (-y, x)$; $180^\circ$: $(x, y) \to (-x, -y)$; $270^\circ$: $(x, y) \to (y, -x)$.
- A $90^\circ$ clockwise turn is the same as a $270^\circ$ counterclockwise turn.
- The center does not move. Every point stays the same distance from the center.
- Example: $(2, 5)$ rotated $90^\circ$ counterclockwise becomes $(-5, 2)$.
Practice
What is the image of $(2, 5)$ after a $90^\circ$ counterclockwise rotation about the origin?
The rule is $(x, y) \to (-y, x)$.
What is the image of $(-3, 4)$ after a $180^\circ$ rotation about the origin?
A $180^\circ$ rotation changes both signs.
What is the image of $(4, -1)$ after a $270^\circ$ counterclockwise rotation about the origin?
The rule is $(x, y) \to (y, -x)$, so $(4, -1) \to (-1, -4)$.
Ready for more? Take the Unit 8 test.