Rotations

A rotation turns a figure around a fixed point. In pre-algebra that point is the origin, $(0, 0)$ , and the turns you need are 90° counterclockwise and 180°. Like translations and reflections, a rotation keeps the figure the same size and shape — it only changes which way the figure points.

Each rotation has a coordinate rule, so you never have to picture the spin in your head. Apply the rule to each vertex, plot the images, and connect them.

The rotation rules

90° counterclockwise: $(x, y) \to (-y, x)$ . The coordinates swap places, and the new first coordinate changes sign. So $(4, 1)$ rotates to $(-1, 4)$ .

180°: $(x, y) \to (-x, -y)$ . The coordinates stay in place, but both change sign. So $(4, 1)$ rotates to $(-4, -1)$ . A 180° turn lands in the same spot whether you spin clockwise or counterclockwise, so no direction is needed.

A quick way to keep them straight: 90° swaps, 180° does not. If the coordinates traded places, a quarter turn happened; if they only changed signs, it was a half turn.

Telling rotations apart on a graph

Compare one vertex of the original figure with its matching image vertex. If both coordinates changed sign and stayed in the same order, the figure was rotated 180°. If the coordinates swapped places, it was a 90° rotation.

In the graph below, the black triangle's corner at $(1, 1)$ matches the blue corner at $(-1, -1)$ . Both signs changed and nothing swapped, so this is a 180° rotation about the origin.

What a rotation preserves

The image of a rotation is congruent to the original: side lengths and angles are unchanged. Every point also stays the same distance from the origin — it just travels around it, like a seat on a Ferris wheel.

The origin itself never moves. If a vertex of your figure is at $(0, 0)$ , its image is still $(0, 0)$ after any rotation about the origin.

Worked examples

Example 1: rotate a point 90° counterclockwise

Find the image of $(6, 2)$ after a 90° counterclockwise rotation about the origin.

Write the 90° counterclockwise rule

(x, y) \to (-y, x)

Swap the coordinates

(2, 6)

Change the sign of the new first coordinate

(-2, 6)

Answer: $(-2, 6)$

Example 2: rotate a point 180°

Find the image of $(-3, 5)$ after a 180° rotation about the origin.

Write the 180° rule

(x, y) \to (-x, -y)

Change the sign of

x

-(-3) = 3

Change the sign of

y

-(5) = -5

Answer: $(3, -5)$

Example 3: rotate a triangle 90° counterclockwise

Rotate triangle $ABC$ with $A(2, 1)$ , $B(4, 1)$ , $C(4, 3)$ 90° counterclockwise about the origin.

The rule for each vertex

(x, y) \to (-y, x)

Apply the rule to

A(2, 1)

A' = (-1, 2)

Apply the rule to

B(4, 1)

B' = (-1, 4)

Apply the rule to

C(4, 3)

C' = (-3, 4)

Answer: $A'(-1, 2)$ , $B'(-1, 4)$ , $C'(-3, 4)$

Try one yourself

Common questions

What is the rule for 90° clockwise?

It is $(x, y) \to (y, -x)$ — the coordinates swap, and the new second coordinate changes sign. A 90° clockwise turn lands in the same place as a 270° counterclockwise turn.

Does it matter which direction I rotate for 180°?

No. Half a turn clockwise and half a turn counterclockwise end at the same spot, so the 180° rule $(x, y) \to (-x, -y)$ has no direction attached.

How do I remember the 90° counterclockwise rule?

Swap, then flip the front: the coordinates trade places, and the new first coordinate changes sign. Test it on an easy point — $(1, 0)$ sits on the positive $x$ -axis, and a quarter turn counterclockwise should carry it to $(0, 1)$ on the positive $y$ -axis. The rule agrees.

Does a rotation change the size of the figure?

No. Rotations, reflections, and translations all produce congruent images. Only a dilation changes size.

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Rotations

The rotation rules

Telling rotations apart on a graph

What a rotation preserves

Worked examples

Example 1: rotate a point 90° counterclockwise

Example 2: rotate a point 180°

Example 3: rotate a triangle 90° counterclockwise

Try one yourself

Common questions

Keep learning

Translations

Reflections

Dilations