Math Core

A rotation turns a figure around a fixed point, the way a wheel turns around its axle. You may already know the rules for turning about the origin. In this lesson you'll see why they work, and use that understanding to rotate about any point and to find a rotation from a figure and its image.

Describing a rotation

Definition

Rotation

A rotation of $\theta$ degrees about a point $P$ (the center of rotation) maps every point $Q$ to a point $Q'$ so that $PQ' = PQ$ and $m\angle QPQ' = \theta$ . The center maps to itself.

A rotation needs three pieces of information: the center, the angle, and the direction. Positive angles are counterclockwise unless a problem says otherwise, so a rotation of $-90^\circ$ is $90^\circ$ clockwise. Because a full turn is $360^\circ$ , a $270^\circ$ counterclockwise rotation lands in the same place as a $90^\circ$ clockwise one.

Rules about the origin

Picture the segment from the origin to $(x, y)$ as the diagonal of a rectangle that is $x$ wide and $y$ tall. A quarter turn counterclockwise tips that rectangle onto its side: it becomes $y$ wide and $x$ tall, and it now reaches to the left. That's why the coordinates swap and the new x-coordinate is $-y$ .

Rotations about the origin

rotation	rule	example with $(3, 1)$
$90^\circ$ counterclockwise	$(x, y) \rightarrow (-y, x)$	$(-1, 3)$
$180^\circ$	$(x, y) \rightarrow (-x, -y)$	$(-3, -1)$
$270^\circ$ counterclockwise ( $90^\circ$ clockwise)	$(x, y) \rightarrow (y, -x)$	$(1, -3)$

Notice that applying the $90^\circ$ rule twice gives $(x, y) \rightarrow (-y, x) \rightarrow (-x, -y)$ , the $180^\circ$ rule, exactly as it should.

Worked example: A quarter turn clockwise

Rotate triangle $ABC$ with $A(1, 1)$ , $B(4, 2)$ and $C(2, 4)$ $90^\circ$ clockwise about the origin.

Use $(x, y) \rightarrow (y, -x)$ :

\begin{aligned} A(1, 1) &\rightarrow A'(1, -1) \\ B(4, 2) &\rightarrow B'(2, -4) \\ C(2, 4) &\rightarrow C'(4, -2) \end{aligned}

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

The triangle moved from Quadrant I to Quadrant IV, one quarter turn clockwise. ✓

Common mistake

After swapping the coordinates, it's easy to negate the wrong one. Check the quadrant: a $90^\circ$ counterclockwise turn moves Quadrant I to II, II to III, III to IV and IV to I. Clockwise goes the other way. If your image is in the wrong quadrant, you negated the wrong coordinate.

Rotating about any point

The origin rules only work when the center is the origin. For a different center $P(h, k)$ , use a three-step plan: shift, rotate, shift back.

Find the position of the point relative to the center: subtract $h$ and $k$ .
Rotate that relative position with the origin rule.
Add $h$ and $k$ back.

Rotating about (h, k)

To rotate $Q(x, y)$ about $P(h, k)$ : compute $(x - h,\ y - k)$ , apply the origin rule for the angle, then add $(h, k)$ to the result.

Worked example: Rotating a triangle about (1, 2)

Rotate triangle $ABC$ with $A(3, 2)$ , $B(5, 2)$ and $C(3, 5)$ $90^\circ$ counterclockwise about $P(1, 2)$ .

Work with positions relative to $P$ , rotate with $(x, y) \rightarrow (-y, x)$ , then add $(1, 2)$ :

vertex	relative to $P$	rotated	image
$A(3, 2)$	$(2, 0)$	$(0, 2)$	$A'(1, 4)$
$B(5, 2)$	$(4, 0)$	$(0, 4)$	$B'(1, 6)$
$C(3, 5)$	$(2, 3)$	$(-3, 2)$	$C'(-2, 4)$

Triangle ABC rotated 90° counterclockwise about P(1, 2).Open in grapher →

Check one vertex: $PA = 2$ and $PA' = 2$ , and $\overline{PA}$ points right while $\overline{PA'}$ points up, a quarter turn counterclockwise. ✓

Finding the rotation

Sometimes you're given a figure and its image and asked which rotation it was. Compare a vertex with its image and test the rules.

Worked example: Which rotation?

A rotation about the origin maps $A(-2, 5)$ to $A'(5, 2)$ . What is the rotation?

Try the rules on $(-2, 5)$ :

$90^\circ$ counterclockwise: $(-5, -2)$ . No.
$180^\circ$ : $(2, -5)$ . No.
$90^\circ$ clockwise: $(5, 2)$ . Yes.

So it's a $90^\circ$ clockwise rotation, which is the same as $270^\circ$ counterclockwise. The quadrants agree: II to I is a quarter turn clockwise.

For a $180^\circ$ rotation there's an especially quick fact: the center is the midpoint of every point and its image, because $Q$ , $P$ and $Q'$ lie on one line with $PQ = PQ'$ .

What a rotation preserves

A rotation is a rigid motion, so it preserves distances and angle measures and the image is congruent to the pre-image. Unlike a reflection, a rotation preserves orientation: if $A \to B \to C$ goes counterclockwise, so does $A' \to B' \to C'$ .

Tip

Angles bigger than $360^\circ$ or negative angles are fine: add or subtract $360^\circ$ until the angle is between $0^\circ$ and $360^\circ$ . For example, $450^\circ$ is the same as $90^\circ$ , and $-90^\circ$ is the same as $270^\circ$ .

Practice

Practice 1

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

Enter a point like (2, -3)

Practice 2

Rotate the point $(-2, -7)$ $270^\circ$ counterclockwise about the origin. What is the image?

Enter a point like (2, -3)

Practice 3

Rotate the point $(3, -8)$ $180^\circ$ about the origin. What is the image?

Enter a point like (2, -3)

Practice 4

A rotation about the origin maps $A(1, 3)$ to $A'(-3, 1)$ . What is the angle of rotation, in degrees counterclockwise, between $0^\circ$ and $360^\circ$ ?

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

Rotate the point $(-1, 4)$ $180^\circ$ about the point $(2, 1)$ . What is the image?

Enter a point like (2, -3)

Practice 6

Rotate the point $(5, 1)$ $90^\circ$ clockwise about the point $(2, 3)$ . What is the image?

Enter a point like (2, -3)

Practice 7

A $180^\circ$ rotation maps $(1, 5)$ to $(7, -1)$ . What is the center of rotation?

Enter a point like (2, -3)

Practice 8

Which rotation about the origin has the same effect as a rotation of $-630^\circ$ ?