Math Core

A Ferris wheel car, the hands of a clock and a spinning fan blade all move the same way: they turn around a fixed center. In geometry, that turn is called a rotation. On the coordinate plane, rotations around the origin follow a few neat coordinate rules.

What a rotation is

Definition

Rotation

A rotation turns every point of a figure through the same angle around a fixed point called the center of rotation. Each point stays the same distance from the center.

To describe a rotation, you need three things: the center, the angle, and the direction. Unless a problem says otherwise, a rotation is counterclockwise (the opposite way a clock's hands move). In this lesson, the center is always the origin, $(0, 0)$ .

The coordinate rules

When you rotate a point $90^\circ$ , its coordinates swap places and one of them changes sign. When you rotate $180^\circ$ , both coordinates change sign.

Rotations about the origin

rotation	rule	example with $(4, 1)$
$90^\circ$ counterclockwise	$(x, y) \rightarrow (-y, x)$	$(-1, 4)$
$180^\circ$	$(x, y) \rightarrow (-x, -y)$	$(-4, -1)$
$90^\circ$ clockwise	$(x, y) \rightarrow (y, -x)$	$(1, -4)$

A $90^\circ$ clockwise turn ends in the same place as a $270^\circ$ counterclockwise turn. And a $180^\circ$ turn lands in the same spot whichever way you go.

Worked example: A quarter turn counterclockwise

Rotate triangle $ABC$ with $A(1, 1)$ , $B(4, 1)$ and $C(4, 3)$ $90^\circ$ counterclockwise about the origin.

Use $(x, y) \rightarrow (-y, x)$ : the new x-coordinate is the opposite of the old y, and the new y-coordinate is the old x.

\begin{aligned} A(1, 1) &\rightarrow A'(-1, 1) \\ B(4, 1) &\rightarrow B'(-1, 4) \\ C(4, 3) &\rightarrow C'(-3, 4) \end{aligned}

Triangle ABC rotated 90° counterclockwise about the origin.Open in grapher →

The triangle started in Quadrant I and a quarter turn counterclockwise carried it into Quadrant II. Side $AB$ was horizontal, and $A'B'$ is vertical.

Worked example: A half turn

Rotate triangle $ABC$ with $A(2, 1)$ , $B(5, 2)$ and $C(3, 4)$ $180^\circ$ about the origin.

Use $(x, y) \rightarrow (-x, -y)$ : $A'(-2, -1)$ , $B'(-5, -2)$ and $C'(-3, -4)$ .

Triangle ABC rotated 180° about the origin.Open in grapher →

The image is in the opposite quadrant. It looks upside down, but it has not been flipped over like a reflection.

Worked example: Clockwise

Rotate the point $P(-3, 5)$ $90^\circ$ clockwise about the origin.

Use $(x, y) \rightarrow (y, -x)$ . The new x-coordinate is the old y, which is $5$ . The new y-coordinate is the opposite of the old x, which is $-(-3) = 3$ . So $P' = (5, 3)$ .

Check with quadrants: $P$ is in Quadrant II. A quarter turn clockwise moves Quadrant II to Quadrant I, and $(5, 3)$ is in Quadrant I. ✓

Common mistake

The $90^\circ$ rules are easy to mix up. After you swap the coordinates, make sure you negate the right one. A quick check: a $90^\circ$ counterclockwise turn moves Quadrant I → II → III → IV → I, and a clockwise turn goes the other way. If your answer lands in the wrong quadrant, you negated the wrong coordinate.

What stays the same

A rotation is a rigid motion. Lengths and angle measures stay the same, so the image is an exact copy. Unlike a reflection, a rotation also keeps orientation: if $A \to B \to C$ goes counterclockwise, so does $A' \to B' \to C'$ .

Tip

If you can't remember a rule, turn your paper. Plot the point, rotate the whole page a quarter turn, and read where the point ended up against the axes.

Practice

Practice 1

Rotate the point $(4, -7)$ $180^\circ$ about the origin. What is the image?

Enter a point like (2, -3)

Practice 2

Rotate the point $(2, 5)$ $90^\circ$ counterclockwise about the origin. What is the image?

Enter a point like (2, -3)

Practice 3

Rotate the point $(2, 5)$ $90^\circ$ clockwise about the origin. What is the image?

Enter a point like (2, -3)

Practice 4

A $270^\circ$ counterclockwise rotation about the origin gives the same image as which transformation?

Practice 5

Rotate the point $(-6, -1)$ $90^\circ$ counterclockwise about the origin. What is the image?

Enter a point like (2, -3)

Triangle ABC and its image A′B′C′.Open in grapher →

Practice 6

Which rotation about the origin maps triangle $ABC$ onto triangle $A'B'C'$ in the graph above?

Practice 7

A point is rotated $90^\circ$ counterclockwise about the origin, and its image is $(-2, 7)$ . What was the original point?

Enter a point like (2, -3)