A rotation turns a figure about a point, the center, by an angle. Like reflections, it keeps size and shape. Here the center is the origin and the turns are quarter and half turns.
Why it works
A point $(x, y)$ is $x$ steps right and $y$ steps up from the origin. Turn the whole picture a quarter turn counterclockwise: “right” becomes “up” and “up” becomes “left.” So the point ends up $x$ steps up and $y$ steps left, which is $(-y, x)$. Check with $(1, 0)$: a quarter turn counterclockwise takes it to $(0, 1)$, and the rule agrees.
| Turn about the origin | Rule |
|---|---|
| $90^\circ$ counterclockwise | $(x, y) \to (-y, x)$ |
| $180^\circ$ | $(x, y) \to (-x, -y)$ |
| $90^\circ$ clockwise | $(x, y) \to (y, -x)$ |
Key idea
To rotate about the origin, use the rule for the angle and direction. A $90^\circ$ clockwise turn is the same as $270^\circ$ counterclockwise. When unsure which rule is which, test it on $(1, 0)$: counterclockwise sends it to $(0, 1)$, a half turn to $(-1, 0)$, clockwise to $(0, -1)$.
Worked examples
Worked example: A quarter turn counterclockwise
Rotate $(4, 1)$ by $90^\circ$ counterclockwise about the origin.
- Rule $(x, y) \to (-y, x)$.
- $(4, 1) \to (-1, 4)$.
Answer: $(-1, 4)$
Worked example: A quarter turn clockwise
Rotate $(4, 1)$ by $90^\circ$ clockwise about the origin.
- Rule $(x, y) \to (y, -x)$.
- $(4, 1) \to (1, -4)$. The point moved from the first quadrant down into the fourth.
Answer: $(1, -4)$
Worked example: A half turn
Rotate $(-2, 5)$ by $180^\circ$ about the origin.
- Both coordinates change sign.
Answer: $(2, -5)$
Common mistake
Mixing up the clockwise and counterclockwise rules. Test the rule on $(1, 0)$: counterclockwise should send it up to $(0, 1)$; clockwise should send it down to $(0, -1)$.