Rotations

A transformation moves or resizes a figure on the coordinate plane. The original figure is the pre-image, and the result is the image, marked with prime notation: point $A$ maps to $A'$ . There are four transformations to know: translations slide, reflections flip, rotations turn, and dilations resize.

Each one comes with a coordinate rule — a recipe like $(x, y) \to (x + 3, y - 5)$ that tells you exactly where every point lands. Once you know the rules, any transformation problem is just plugging in coordinates one vertex at a time.

The four transformations

A translation slides every point the same distance in the same direction — nothing turns or flips. A reflection flips the figure across a line (the line of reflection), producing a mirror image. A rotation turns the figure around a fixed point (usually the origin) by some angle. A dilation stretches or shrinks the figure away from or toward a center point by a scale factor.

The figure below shows a translation: $\triangle ABC$ slides $5$ units left and $4$ units down to $\triangle A'B'C'$ . Every vertex moves by the same amounts, so the image is the same size and shape as the pre-image.

The coordinate rules

Translations: $(x, y) \to (x + a, y + b)$ , where $a$ is the horizontal move (positive means right) and $b$ is the vertical move (positive means up).

Reflections: across the $x$ -axis, $(x, y) \to (x, -y)$ ; across the $y$ -axis, $(x, y) \to (-x, y)$ ; across the line $y = x$ , $(x, y) \to (y, x)$ . Each axis reflection changes exactly one sign, and the $y = x$ reflection swaps the coordinates.

Rotations about the origin: $90^\circ$ counterclockwise, $(x, y) \to (-y, x)$ ; $180^\circ$ , $(x, y) \to (-x, -y)$ ; $90^\circ$ clockwise, $(x, y) \to (y, -x)$ . Note that a $90^\circ$ rotation both swaps the coordinates and changes one sign — swapping alone is not enough.

Dilations centered at the origin with scale factor $k$ : $(x, y) \to (kx, ky)$ . A scale factor greater than $1$ enlarges the figure; a factor between $0$ and $1$ shrinks it.

Rigid motions vs. dilations

Translations, reflections, and rotations are rigid motions: they preserve every distance and every angle, so the image is congruent to the pre-image. A dilation preserves angles but multiplies every length by the scale factor, so the image is similar to the pre-image — same shape, different size (unless $k = 1$ ).

This is the fastest way to answer which transformation happened questions. If the image is a different size, it involved a dilation. If it is the same size but flipped in orientation, look for a reflection. Same size and orientation, just moved — a translation. Same size but turned — a rotation.

Worked examples

Example 1: a translation

Apply the translation $(x, y) \to (x + 2, y - 3)$ to $\triangle ABC$ with $A(1, 4)$ , $B(3, 0)$ , and $C(-2, 1)$ .

Apply the rule to

A

: add

2

x

, subtract

3

from

y

A(1, 4) \to A'(3, 1)

Apply the rule to

B

B(3, 0) \to B'(5, -3)

Apply the rule to

C

C(-2, 1) \to C'(0, -2)

Answer: $A'(3, 1)$ , $B'(5, -3)$ , $C'(0, -2)$

Example 2: a reflection across the y-axis

Reflect $\triangle ABC$ with $A(2, 1)$ , $B(5, 1)$ , and $C(2, 4)$ across the $y$ -axis.

The rule for a reflection across the

y

-axis changes the sign of

x

only

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

Apply it to

A

A(2, 1) \to A'(-2, 1)

Apply it to

B

B(5, 1) \to B'(-5, 1)

Apply it to

C

C(2, 4) \to C'(-2, 4)

Answer: $A'(-2, 1)$ , $B'(-5, 1)$ , $C'(-2, 4)$

Example 3: a 90-degree counterclockwise rotation

Rotate $\triangle ABC$ with $A(3, 2)$ , $B(5, -1)$ , and $C(0, 4)$ by $90^\circ$ counterclockwise about the origin.

The rule: the new

x

is the opposite of the old

y

, and the new

y

is the old

x

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

Apply it to

A

A(3, 2) \to A'(-2, 3)

Apply it to

B

— the opposite of

-1

1

B(5, -1) \to B'(1, 5)

Apply it to

C

C(0, 4) \to C'(-4, 0)

Answer: $A'(-2, 3)$ , $B'(1, 5)$ , $C'(-4, 0)$

Example 4: a dilation

Dilate the segment with endpoints $A(2, -1)$ and $B(-1, 2)$ by a scale factor of $3$ centered at the origin.

The rule multiplies both coordinates by the scale factor

(x, y) \to (3x, 3y)

Apply it to

A

A(2, -1) \to A'(6, -3)

Apply it to

B

B(-1, 2) \to B'(-3, 6)

The image segment is

3

times as long as the pre-image — similar, not congruent

Answer: $A'(6, -3)$ , $B'(-3, 6)$

Try one yourself

Common questions

How do I keep the two 90-degree rotation rules straight?

Counterclockwise is $(x, y) \to (-y, x)$ and clockwise is $(x, y) \to (y, -x)$ . If you blank on which is which, test one easy point: rotating $(1, 0)$ by $90^\circ$ counterclockwise must land on $(0, 1)$ , straight up. Only the counterclockwise rule does that.

Which transformations keep the figure congruent to the original?

Translations, reflections, and rotations — the rigid motions. They preserve all side lengths and angle measures. A dilation preserves angles but scales lengths, so its image is similar instead, unless the scale factor is $1$ .

What does the prime mark on a point mean?

$A'$ is the image of point $A$ after a transformation. If you apply a second transformation, the new image is $A''$ . The mark keeps the pre-image and image vertices matched up, which is exactly what you need when writing or checking a rule.

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Rotations

The four transformations

The coordinate rules

Rigid motions vs. dilations

Worked examples

Example 1: a translation

Example 2: a reflection across the y-axis

Example 3: a 90-degree counterclockwise rotation

Example 4: a dilation

Try one yourself

Common questions

Keep learning

SSS & SAS Triangle Congruence

AA Similarity

The Distance & Midpoint Formulas on the Coordinate Plane