Rotations

A rotation is a transformation that turns a figure about a fixed point, called the , through a given angle. Every point of the figure travels along a circular arc centered at that point, so the distance from each point to the center never changes. Counter-clockwise angles are taken as positive, following the standard mathematical convention.

Rotating about the origin

Need a refresher? Negative numbers

Picture the number line: positives to the right of zero, negatives to the left.

Adding moves right: −3 + 5 = 2 (start at −3, take 5 steps right).
Subtracting moves left: −3 − 4 = −7 (start at −3, take 4 steps left).
Subtracting a negative flips to adding: 5 − (−2) = 5 + 2 = 7.

A sign flip turns a number into its mirror image across zero: the opposite of 3 is −3, and the opposite of −3 is 3.

Try it: What is −2 − 6?

Make a prediction: The point (3, 0) sits on the positive x-axis. Rotate it 90° counter-clockwise about the origin. Where does it land?

(0, 3)(0, −3)(−3, 0)(3, 0)

For the common quarter-turns about the origin, a rotation by an angle $\theta$ has clean coordinate rules. A $90^\circ$ counter-clockwise rotation sends a point to:

(x, y) \mapsto (-y,\, x)

The half-turn $180^\circ$ sends $(x, y) \mapsto (-x, -y)$ , and the three-quarter turn $270^\circ$ (equivalently $-90^\circ$ ) sends $(x, y) \mapsto (y, -x)$ .

Worked example: Rotate (2, 5) by 90° counter-clockwise about the origin

Pick the rule for a 90° counter-clockwise rotation about the origin.
Before the next step: Which rule applies?
(x, y) ↦ (−y, x)(x, y) ↦ (y, −x)(x, y) ↦ (−x, −y)

Rotating about any center

To rotate about an arbitrary center $(a, b)$ instead of the origin, first translate the figure so the center sits at the origin, apply the rotation above, then translate back. Because a rotation preserves distances and angles, the image is always congruent to the original figure.

Rotations

Rotating about the origin

Rotating about any center

Practice