How do I remember the coordinate rules without mixing them up?

A good way is to think about what happens to the point (1, 0) on the positive x-axis. Under 90° counterclockwise, it lands at (0, 1) on the positive y-axis. Under 180°, it lands at (-1, 0) on the negative x-axis. Under 270° counterclockwise, it lands at (0, -1) on the negative y-axis. You can also picture a point rotating around the origin: the rule swaps and negates coordinates in a specific pattern depending on the angle. Practice writing out the rules several times and testing them on a few points until they stick.

Is a 270° counterclockwise rotation the same as a 90° clockwise rotation?

Yes, they are equivalent. A 270° counterclockwise turn ends up in the same place as a 90° clockwise turn. However, in this course we use the counterclockwise convention, so always describe rotations as counterclockwise angles. If you ever encounter a clockwise rotation in a problem, convert it: a 90° clockwise rotation is the same as a 270° counterclockwise rotation, a 180° rotation is the same either way, and a 270° clockwise rotation is the same as a 90° counterclockwise.

What if the center of rotation is not the origin?

This lesson focuses only on rotations about the origin. If the center were a different point, the coordinate rules would be different and the process would be more complex. You would first translate so that the center moves to the origin, then apply the rotation rule, then translate back. For now, assume the center is always the origin unless told otherwise.

Why is distance from the origin preserved during a rotation?

When you rotate a point about the origin, it moves along a circular arc centered at the origin. A circle is defined as the set of all points at a fixed distance from the center, so every point on that circle is the same distance from the origin. Rotating the point doesn't move it closer to or farther from the origin—it just moves it around the circle. This is why the distance is always preserved, and it's the geometric reason why the distance between any two points in the figure is also preserved.

M8MATH-8.2

Rotations About the Origin

Learn how to rotate figures 90°, 180°, and 270° about the origin using coordinate rules, and identify rotations from preimages and images.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Rotations About the Origin, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

Start Rotations About the Origin free

What this lesson covers

A rotation is a turn of a figure around a fixed point. In this lesson, we focus on rotations about the origin of a coordinate plane. You'll discover the coordinate rules that describe where each point moves, practice rotating whole figures, and verify that distances and angles stay the same. These skills help you understand how shapes move in space and prepare you for more complex transformations.

Understanding Rotations About the Origin

A rotation is a rigid motion that turns a figure around a center point. When the center is the origin (0, 0), we measure rotation using degrees and a direction. By convention, we measure angles counterclockwise from the positive x-axis. A 90° counterclockwise rotation turns a figure one-quarter turn to the left. A 180° rotation is a half-turn, and a 270° rotation is three-quarter turn. The key property of a rotation is that it preserves distance: every point stays the same distance from the center of rotation, and the figure itself doesn't change shape or size. This makes rotation a rigid motion, just like translation and reflection.

To rotate a figure about the origin, we apply a coordinate rule to each vertex. For a 90° counterclockwise rotation, the rule is

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

. For 180°, the rule is

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

. For 270° counterclockwise (or equivalently, 90° clockwise), the rule is

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

. These rules come from the geometry of rotation: when you rotate a point counterclockwise around the origin, its x-coordinate and y-coordinate swap and change signs in a predictable pattern.

The Three Coordinate Rules

Each rotation angle has its own coordinate rule. Here is how to apply them:

Rotation	Coordinate Rule	What Happens
90° counterclockwise	$(x, y) \to (-y, x)$	New x is the negative old y; new y is the old x
180°	$(x, y) \to (-x, -y)$	Both coordinates flip sign
270° counterclockwise	$(x, y) \to (y, -x)$	New x is the old y; new y is the negative old x

For example, if point

A = (3, 2)

is rotated 90° counterclockwise about the origin, apply the rule: new

x = -2

, new

y = 3

, so

A' = (-2, 3)

. If you rotate the same point 180°, you get

A' = (-3, -2)

. If you rotate 270° counterclockwise, you get

A' = (2, -3)

. Notice that rotating 270° counterclockwise gives the same result as rotating 90° clockwise—both are equivalent. To use these rules on a figure, apply the rule to every vertex, then connect the new vertices in the same order as the original.

Why Distances and Angles Are Preserved

A rotation is a rigid motion, which means the image is congruent to the preimage. Two key properties are always preserved: distances between points and angle measures within the figure.

Distances are preserved because every point rotates around the origin at the same angle, staying the same distance from the origin. If two points are, say, 5 units apart before a rotation, they will be 5 units apart after. You can verify this using the distance formula on the original points and the rotated points.

Angles are preserved because rotation turns the entire figure uniformly. If a corner of the preimage has a 60° angle, the corresponding corner in the image will also be 60°. This is a consequence of rotating every point the same amount: the shape doesn't stretch, shrink, or distort.

Because these properties hold, a rotated figure is always congruent to the original. This is why rotations are called rigid motions: they move figures without changing their shape or size.

Identifying a Rotation from Preimage and Image

Sometimes you're given a preimage and an image and asked to determine if they are related by a rotation, and if so, which one. To identify a rotation, follow these steps. First, check that the preimage and image are congruent: corresponding sides should be equal and corresponding angles should be equal. If they are not congruent, no rotation relates them. Second, pick one vertex from the preimage and its corresponding image point. Plot both on the coordinate plane and see what angle would rotate the first to the second around the origin. Third, verify your answer by checking whether the same rotation moves another vertex correctly. If all vertices are rotated by the same angle in the same direction (counterclockwise), then a rotation has taken place.

A common place students go wrong is assuming that any shape in a different place is a rotation. Reflections and translations also move figures. The key difference is that rotation causes points to move in a circular arc around the origin, not in a straight line (translation) or across a line of reflection. You can test this by checking whether the angle from the origin to any point has increased by the same amount.

Worked Example: Rotating a Triangle

Problem: Triangle

PQR

has vertices

P = (4, 1)

Q = (4, 3)

, and

R = (1, 3)

. Rotate it 90° counterclockwise about the origin. Give the coordinates of the image

P'Q'R'

and verify that side

PQ

has the same length as side

P'Q'

.

Solution: Apply the rule

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

to each vertex.

For

P = (4, 1)

: new

x = -1

, new

y = 4

, so

P' = (-1, 4)

.

For

Q = (4, 3)

: new

x = -3

, new

y = 4

, so

Q' = (-3, 4)

.

For

R = (1, 3)

: new

x = -3

, new

y = 1

, so

R' = (-3, 1)

.

The image triangle has vertices

P' = (-1, 4)

Q' = (-3, 4)

R' = (-3, 1)

.

To verify that distance is preserved, find the length of

PQ

in the preimage:

PQ = \sqrt{(4-4)^2 + (3-1)^2} = \sqrt{0 + 4} = 2

Now find the length of

P'Q'

in the image:

P'Q' = \sqrt{(-1-(-3))^2 + (4-4)^2} = \sqrt{4 + 0} = 2

Both segments have length 2, confirming that distance is preserved under rotation.

Key terms

Rotation.: A rigid motion that turns a figure around a fixed center point by a given angle, measured counterclockwise by convention.
Center of rotation.: The fixed point around which a figure is turned. In this lesson, the center is always the origin (0, 0).
Counterclockwise.: The direction of rotation measured counterclockwise from the positive x-axis, which is the standard convention in mathematics.
Preimage.: The original figure before a transformation is applied.
Image.: The figure that results after a transformation is applied.
Rigid motion.: A transformation that preserves distance and angle measure, so the image is congruent to the preimage.
Congruent.: Two figures are congruent if they have the same shape and size; one can be moved to match the other exactly through a rigid motion.
Coordinate rule.: A formula such as $(x, y) \to (-y, x)$ that describes how the coordinates of each point change under a transformation.

Worked example

Point

M = (2, 5)

is rotated 180° about the origin. Find the coordinates of

M'

. Then rotate

M'

by 90° counterclockwise to find

M''

. What is

M''

Start with

M = (2, 5)

and apply the 180° rule

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

.

For the first rotation:

M' = (-2, -5)

.

Now take

M' = (-2, -5)

and apply the 90° counterclockwise rule

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

.

For the second rotation: new

x = -(-5) = 5

, new

y = -2

, so

M'' = (5, -2)

.

You can verify this another way: rotating 180° and then 90° counterclockwise is the same as rotating

180° + 90° = 270°

counterclockwise. Applying the 270° rule directly to

M = (2, 5)

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

gives

(5, -2)

. Both methods give the same answer, which is expected because a series of rotations about the same center is equivalent to a single rotation by the sum of the angles.

Practice questions

Point

A = (6, 2)

is rotated 90° counterclockwise about the origin. What are the coordinates of

A'

$(-2, 6)$
$(2, 6)$
$(-6, -2)$
$(6, -2)$

Answer: $(-2, 6)$

Use the 90° counterclockwise rule

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

. Here,

x = 6

and

y = 2

, so the new coordinates are

(-2, 6)

. Many students accidentally swap the coordinates without changing signs, getting

(2, 6)

—a common error. Others may confuse this with the 180° or 270° rule. Always check that you've applied the correct rule for the angle given.

Triangle

ABC

has vertices at

A = (3, 0)

B = (3, 4)

, and

C = (0, 4)

. After a 270° counterclockwise rotation about the origin, the image triangle has vertices at

A' = (0, -3)

B' = (4, -3)

, and

C' = (4, 0)

. Verify that this rotation preserves the distance from

A

C

by calculating both

AC

and

A'C'

Answer: $AC = 5$ and $A'C' = 5$

For the preimage:

AC = \sqrt{(3-0)^2 + (0-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5

. For the image:

A'C' = \sqrt{(0-4)^2 + (-3-0)^2} = \sqrt{16 + 9} = \sqrt{25} = 5

. Both distances equal 5, confirming that the rotation is a rigid motion and preserves distance. If your distances don't match, recheck your coordinate calculations—each vertex must be rotated using the correct rule.

The square with vertices

(1, 0)

(1, 1)

(0, 1)

, and

(0, 0)

is rotated about the origin. The image has vertices at

(0, 1)

(-1, 1)

(-1, 0)

, and

(0, 0)

. Identify the angle of rotation (90°, 180°, or 270° counterclockwise).

Answer: 90° counterclockwise

Track one vertex that is not at the origin. The vertex

(1, 0)

maps to

(0, 1)

. Apply each rule to

(1, 0)

: the 90° rule

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

gives

(0, 1)

✓, the 180° rule gives

(-1, 0)

, and the 270° rule gives

(0, -1)

. Only the 90° rule matches, so the rotation is 90° counterclockwise. To be thorough, you can check another vertex:

(1, 1) \to (-1, 1)

✓ under the 90° rule.

FAQ

How do I remember the coordinate rules without mixing them up?: A good way is to think about what happens to the point $(1, 0)$ on the positive x-axis. Under 90° counterclockwise, it lands at $(0, 1)$ on the positive y-axis. Under 180°, it lands at $(-1, 0)$ on the negative x-axis. Under 270° counterclockwise, it lands at $(0, -1)$ on the negative y-axis. You can also picture a point rotating around the origin: the rule swaps and negates coordinates in a specific pattern depending on the angle. Practice writing out the rules several times and testing them on a few points until they stick.
Is a 270° counterclockwise rotation the same as a 90° clockwise rotation?: Yes, they are equivalent. A 270° counterclockwise turn ends up in the same place as a 90° clockwise turn. However, in this course we use the counterclockwise convention, so always describe rotations as counterclockwise angles. If you ever encounter a clockwise rotation in a problem, convert it: a 90° clockwise rotation is the same as a 270° counterclockwise rotation, a 180° rotation is the same either way, and a 270° clockwise rotation is the same as a 90° counterclockwise.
What if the center of rotation is not the origin?: This lesson focuses only on rotations about the origin. If the center were a different point, the coordinate rules would be different and the process would be more complex. You would first translate so that the center moves to the origin, then apply the rotation rule, then translate back. For now, assume the center is always the origin unless told otherwise.
Why is distance from the origin preserved during a rotation?: When you rotate a point about the origin, it moves along a circular arc centered at the origin. A circle is defined as the set of all points at a fixed distance from the center, so every point on that circle is the same distance from the origin. Rotating the point doesn't move it closer to or farther from the origin—it just moves it around the circle. This is why the distance is always preserved, and it's the geometric reason why the distance between any two points in the figure is also preserved.

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