Transformations: Translations, Reflections and Rotations

Quick answer

A translation slides a figure, a reflection flips it across a line, and a rotation turns it about a point. All three are rigid motions: they move a figure without resizing or reshaping it, so every side length and every angle survives unchanged. Each has a coordinate rule, and reflecting across the y-axis negates the x of every vertex.

What you'll learn

Apply a translation, reflection or rotation to a figure
Write the coordinate rule for each transformation
Explain what a rigid motion preserves

Moving a figure without changing it

A transformation moves a figure. These three move it without altering its size or shape, which makes them rigid motions:

Transformation	What it does
translation	slides
reflection	flips across a line
rotation	turns about a point

The original is the pre-image and the result is the image. A vertex $A$ maps to $A'$ , said “A prime”.

Translation: sliding

Add the same amount to every vertex.

(x, \; y) \;\rightarrow\; (x + a, \; y + b)

A triangle translated 5 right and 2 down

Every vertex moved $5$ right and $2$ down:

(-4, 3) \rightarrow (1, 1) \qquad (-1, 3) \rightarrow (4, 1) \qquad (-4, 1) \rightarrow (1, -1)

Because every vertex moved by the same amount, the distances between them are untouched. The figure arrives identical, in a new place.

Reflection: flipping across a line

Reflect across	Rule
the $y$ -axis	$(x, y) \rightarrow (-x, y)$
the $x$ -axis	$(x, y) \rightarrow (x, -y)$

A shape reflected across the y-axis

The reflecting line is a mirror. Each point and its image sit the same distance from it, on opposite sides.

Note which coordinate changes. Reflecting across the $y$ -axis negates the $x$ , because the $y$ -axis is the vertical mirror and crossing it changes how far left or right a point is. The axis you reflect across is the one that stays put.

Rotation: turning about a point

Rotations here are about the origin, measured counterclockwise unless stated.

Rotation	Rule
$90°$ counterclockwise	$(x, y) \rightarrow (-y, x)$
$180°$	$(x, y) \rightarrow (-x, -y)$
$270°$ counterclockwise	$(x, y) \rightarrow (y, -x)$

A triangle rotated 180 degrees about the origin

A $180°$ rotation negates both coordinates, sending each point straight through the origin to the opposite side.

(1, 1) \rightarrow (-1, -1) \qquad (5, 1) \rightarrow (-5, -1) \qquad (1, 4) \rightarrow (-1, -4)

What all three preserve

This is what the standards ask you to verify, and it is the reason these three are grouped together.

Preserved	Why it matters
lengths of segments	the image is the same size
measures of angles	the image is the same shape
parallel lines stay parallel	the image is not sheared

A segment maps to a segment of equal length. An angle maps to an angle of equal measure. Two parallel lines map to two lines that are still parallel.

Check it on the translation above. The pre-image side from $(-4, 3)$ to $(-1, 3)$ is $3$ units long. Its image runs from $(1, 1)$ to $(4, 1)$ , which is also $3$ . Nothing could have changed it, because both endpoints moved by the identical amount.

Why rigid motions are worth naming

They give congruence a definition you can act on.

Saying two figures are “the same” is vague. Saying one can be carried onto the other by a sequence of translations, reflections and rotations is a test you can actually carry out — and that is exactly how congruence is defined.

It also explains why some figures need more than one move. A shape that is flipped and shifted needs a reflection and a translation, and the pair still counts as a rigid motion because each step preserves everything.

The distinction matters against the one transformation that is not rigid: a dilation resizes a figure, so lengths change while angles do not. That produces similarity rather than congruence, and it is why dilations are handled separately.

Worked examples

Common mistakes

Practice problems

Translate $(3, 4)$ by $2$ right and $5$ down.

Answer

$(5, -1)$

Full solution

$(3 + 2, \; 4 - 5) = (5, -1)$ .
Reflect $(7, 3)$ across the $y$ -axis.

Answer

$(-7, 3)$

Full solution

The $x$ negates and the $y$ stays.
Reflect $(4, -6)$ across the $x$ -axis.

Answer

$(4, 6)$

Full solution

The $y$ negates and the $x$ stays.
Rotate $(5, 2)$ by $180°$ about the origin.

Answer

$(-5, -2)$

Full solution

Both coordinates negate.
Rotate $(1, 3)$ by $90°$ counterclockwise about the origin.

Answer

$(-3, 1)$

Full solution

$(x, y) \rightarrow (-y, x)$ , so $(1, 3) \rightarrow (-3, 1)$ .
Which transformation slides a figure without turning or flipping it?

Answer

A translation

Full solution

It is described only by how far across and how far up or down.
A triangle has a side of length $8$ . After a rotation, how long is that side?

Answer

$8$

Full solution

Rotations are rigid motions, so every length is preserved.
Translate the triangle $(0, 0)$ , $(3, 0)$ , $(0, 4)$ by $1$ left and $2$ up.

Hint

Every vertex moves the same way.

Answer

$(-1, 2)$ , $(2, 2)$ , $(-1, 6)$

Full solution

Subtract $1$ from each $x$ and add $2$ to each $y$ .

The sides remain $3$ and $4$ long, since all three corners moved identically.
A segment from $(2, 1)$ to $(6, 1)$ is reflected across the $y$ -axis. Give the image and its length.

Answer

From $(-2, 1)$ to $(-6, 1)$ , length $4$

Full solution

Each $x$ negates: $(2, 1) \rightarrow (-2, 1)$ and $(6, 1) \rightarrow (-6, 1)$ .

Length: $|-2 - (-6)| = 4$ , the same as the original $|6 - 2| = 4$ .
Reflecting $(5, 2)$ across the $x$ -axis, Jon writes $(-5, 2)$ . Find his error.

Hint

Which axis is the mirror, and which coordinate measures distance from it?

Answer

He negated the $x$ . Across the $x$ -axis it is $(5, -2)$ .

Full solution

The $x$ -axis is a horizontal mirror. What changes when you cross it is how far up or down a point sits, which is the $y$ .

So $(5, 2) \rightarrow (5, -2)$ : two units above becomes two units below, and the horizontal position is untouched.

Jon applied the $y$ -axis rule. A quick check catches it — his image sits on the same side of the $x$ -axis as the original, which no reflection across that axis could produce.

Frequently asked questions

What is a rigid motion?

A transformation that moves a figure without changing its size or shape. Translations, reflections and rotations are all rigid motions.

What is the rule for reflecting across the y-axis?

Negate the x-coordinate and leave y alone, so (x, y) becomes (−x, y). Reflecting across the x-axis does the opposite.

How do I translate a figure?

Add the same amount to every vertex. Moving 3 right and 2 down sends (x, y) to (x + 3, y − 2).

What does a 90 degree rotation about the origin do?

Counterclockwise, (x, y) becomes (−y, x). Clockwise, it becomes (y, −x).

Do transformations change side lengths?

Not these three. Translations, reflections and rotations preserve every length and every angle, which is what makes them rigid.

What to learn next

Standards alignment

This lesson covers the following Common Core State Standards for Mathematics.

CCSS.MATH.CONTENT.8.G.A.1GeometryVerify experimentally the properties of rotations, reflections, and translations:
CCSS.MATH.CONTENT.8.G.A.1aGeometryLines are taken to lines, and line segments to line segments of the same length.
CCSS.MATH.CONTENT.8.G.A.1bGeometryAngles are taken to angles of the same measure.
CCSS.MATH.CONTENT.8.G.A.1cGeometryParallel lines are taken to parallel lines.

For teachersPrint problems onlyTick to print a blank worksheet; leave it clear to print the lesson with the full answer key.

Quick answer

What you'll learn

Frequently asked questions

What is a rigid motion?

What is the rule for reflecting across the y-axis?

How do I translate a figure?

What does a 90 degree rotation about the origin do?

Do transformations change side lengths?

What to learn next

Standards alignment

More Geometry lessons