Rotations

By the end of this lesson, you will be able to rotate figures about the origin by 90°, 180°, and 270° using coordinate rules, apply rotations about other centers, and verify that rotations are isometries that preserve orientation.

The Big Idea

A rotation turns every point of a figure by a fixed angle about a fixed center point. Unlike translations (which slide) and reflections (which flip), rotations spin. They preserve distance, angle, and — unlike reflections — orientation (counterclockwise order stays counterclockwise after a rotation).

Definition

A rotation of $\theta°$ counterclockwise about center $C$ maps every point $P$ to $P'$ such that:

• $CP = CP'$ (equidistant from center)
• $\angle PCP' = \theta°$

Direction convention: positive angles = counterclockwise; negative angles = clockwise.

Coordinate Rules for Rotations About the Origin

Rotation	Rule
90° counterclockwise	$(x, y) \to (-y, x)$
180°	$(x, y) \to (-x, -y)$
270° counterclockwise (= 90° clockwise)	$(x, y) \to (y, -x)$
360°	$(x, y) \to (x, y)$ (identity)

Memory trick for 90° CCW: the new $x$ is $-y$; the new $y$ is the old $x$. Think "negate the first, keep the second, but swap them."

Example 1 — 90° Counterclockwise

Rotate $\triangle ABC$: $A(1, 3)$, $B(4, 3)$, $C(4, 7)$ by 90° CCW about the origin.

$$A' = (-3, 1), \quad B' = (-3, 4), \quad C' = (-7, 4)$$

Example 2 — 180°

Rotate $P(5, -2)$ by 180° about the origin.

$$P' = (-5, 2)$$

Example 3 — 270° (= 90° Clockwise)

Rotate $Q(-3, 4)$ by 270° CCW (equivalently, 90° CW) about the origin.

$$Q' = (4, 3)$$

Using $(y, -x)$: $Q' = (4, -(-3)) = (4, 3)$ ✓

Example 4 — Rotation Not About the Origin

Rotate $P(5, 3)$ by 90° CCW about center $C(2, 1)$.

Translate so $C$ is at origin: $P - C = (3, 2)$

Apply 90° CCW: $(3, 2) \to (-2, 3)$

Translate back: $(-2 + 2, 3 + 1) = (0, 4)$

$P' = (0, 4)$

Rotation Symmetry

A figure has rotational symmetry if it maps onto itself under a rotation of less than 360°.