Why do we say translations and reflections are "rigid" transformations?

Rigid transformations preserve the size and shape of figures. The distances between all pairs of points stay the same, and all angle measures stay the same. This means the image figure is congruent to the original. Translations and reflections both have this property, unlike dilations, which change size.

Do reflections across the x-axis and y-axis use different rules?

Yes. Reflection across the x-axis uses (x, y) → (x, -y): flip the sign of the y-coordinate. Reflection across the y-axis uses (x, y) → (-x, y): flip the sign of the x-coordinate. The axis name tells you which coordinate changes sign—the other one stays the same.

If I translate a figure and then reflect it, do parallel lines stay parallel?

Yes. Each transformation—translation and reflection—preserves parallel lines individually. A translation slides all points the same way, so if two lines were parallel, they still move together and stay parallel. A reflection flips all points consistently across a line, also preserving parallelism. Combining transformations preserves parallelism as well.

How do I know if I applied a coordinate rule correctly?

Substitute the original coordinates of at least two vertices into the rule and plot both the original and image points on a coordinate plane. Check that the movement or flip makes sense (does the figure move the right direction and distance, or is it flipped correctly across the right line?). Also verify that distances or angles look preserved visually—this quick check often catches mistakes in applying the rule.

M8MATH-8.1

Translations & Reflections

Learn how to translate and reflect figures on the coordinate plane using coordinate rules, and verify that rigid transformations preserve distances, angles, and parallel lines.

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

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What this lesson covers

Transformations are ways to move or flip figures on the coordinate plane. In this lesson, you'll learn two fundamental rigid transformations: translations (slides) and reflections (flips). These transformations are "rigid" because they preserve the size and shape of figures — distances and angles stay the same. You'll use coordinate rules to describe exactly how figures move, and you'll verify that important properties like parallel lines and angle measures don't change.

What Is a Translation?

A translation is a transformation that slides a figure from one location to another without rotating or flipping it. Every point on the figure moves the same distance in the same direction. We describe translations using a coordinate rule of the form

(x, y) \to (x + a, y + b)

, where

a

is the horizontal change and

b

is the vertical change.

For example, the rule

(x, y) \to (x + 3, y - 2)

means move every point 3 units right and 2 units down. If a vertex is at

(1, 4)

, it moves to

(1 + 3, 4 - 2) = (4, 2)

. All vertices of the figure follow the same rule, so the entire shape slides as a rigid unit.

Key properties of translations: lines stay lines, angles stay the same size, lengths stay the same, and parallel lines remain parallel. The translated figure is congruent to the original — it has exactly the same dimensions and angles, just in a different position.

What Is a Reflection?

A reflection is a transformation that flips a figure across a line (the line of reflection). The most common reflections in the coordinate plane are across the x-axis and across the y-axis. Reflection across the x-axis uses the rule

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

: the x-coordinate stays the same, and the y-coordinate changes sign. Reflection across the y-axis uses the rule

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

: the y-coordinate stays the same, and the x-coordinate changes sign.

Imagine folding a piece of paper along the x-axis — points above the axis flip down below it, and the distance from the axis is preserved. The same idea applies to reflection across the y-axis. Like translations, reflections preserve lengths, angle measures, and the parallelism of lines. The reflected figure is congruent to the original and is a mirror image across the line of reflection.

Verifying Properties Are Preserved

When we apply a rigid transformation, we need to verify that key geometric properties remain unchanged. For a figure and its image, check these properties:

Lengths: Measure the distance between two points on the original figure using the distance formula

d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}

, then measure the distance between the corresponding points on the transformed figure. They should be equal.

Angle measures: Angles formed by the sides of a figure do not change size under translations or reflections. You can verify this by checking that slopes of corresponding sides stay the same (for translation) or change in a predictable way that preserves angle measures (for reflection).

Parallel lines: If two lines in the original figure are parallel, their images under a translation or reflection are also parallel. This happens because translations move all points the same way, and reflections flip all points consistently across a line.

Lines map to lines: A straight line always maps to a straight line under these rigid transformations. Three collinear points remain collinear in their image.

Applying Coordinate Rules

To transform a figure, apply the coordinate rule to every vertex. If a triangle has vertices at

A(2, 1)

B(5, 1)

, and

C(3, 4)

, and you apply the rule

(x, y) \to (x - 1, y + 3)

, then:

A(2, 1) \to A'(2 - 1, 1 + 3) = A'(1, 4)

B(5, 1) \to B'(5 - 1, 1 + 3) = B'(4, 4)

C(3, 4) \to C'(3 - 1, 4 + 3) = C'(2, 7)

The original and transformed figures are congruent. You can verify this by computing side lengths or angles, or by plotting both figures and observing that one is a rigid movement of the other.

Common mistakes: Students sometimes apply the rule incorrectly by forgetting the

+

or changing

+

-

in the coordinate rule. Always substitute carefully, and double-check at least one vertex to make sure the direction and distance of movement make sense.

Key terms

Translation.: A rigid transformation that slides every point of a figure the same distance in the same direction, described by the rule $(x, y) \to (x + a, y + b)$ .
Reflection.: A rigid transformation that flips a figure across a line of reflection, such as the x-axis or y-axis, creating a mirror image.
Rigid transformation.: A transformation that preserves distances and angle measures, so the image is congruent to the original figure.
Coordinate rule.: A notation such as $(x, y) \to (x + a, y + b)$ that describes how every point of a figure moves during a transformation.
Line of reflection.: The line across which a figure is flipped during a reflection; common examples are the x-axis and y-axis.
Congruent.: Two figures are congruent if they have the same size and shape; one can be obtained from the other by a rigid transformation.

Worked example

Triangle

PQR

has vertices at

P(1, 2)

Q(4, 2)

, and

R(2, 5)

. Apply the translation rule

(x, y) \to (x + 2, y - 1)

to find the image triangle

P'Q'R'

, then verify that the side length

PQ

equals the side length

P'Q'

Step 1: Apply the coordinate rule to each vertex.

P(1, 2) \to P'(1 + 2, 2 - 1) = P'(3, 1)

Q(4, 2) \to Q'(4 + 2, 2 - 1) = Q'(6, 1)

R(2, 5) \to R'(2 + 2, 5 - 1) = R'(4, 4)

Step 2: Find the length of

PQ

in the original triangle.

PQ = \sqrt{(4 - 1)^2 + (2 - 2)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3

Step 3: Find the length of

P'Q'

in the image triangle.

P'Q' = \sqrt{(6 - 3)^2 + (1 - 1)^2} = \sqrt{3^2 + 0^2} = \sqrt{9} = 3

Conclusion:

PQ = P'Q' = 3

, so the translation preserves the length of this side. Because translations are rigid transformations, all side lengths and angle measures are preserved, and triangle

PQR

is congruent to triangle

P'Q'R'

Practice questions

Point

A

is at

(3, -2)

. After a translation, its image

A'

is at

(0, 1)

. Which coordinate rule describes this translation?

$(x, y) \to (x - 3, y + 3)$
$(x, y) \to (x + 3, y + 3)$
$(x, y) \to (x - 3, y - 3)$
$(x, y) \to (x + 3, y - 3)$

Answer: $(x, y) \to (x - 3, y + 3)$

To find the translation rule, subtract the original coordinates from the image coordinates. The change in

x

0 - 3 = -3

, so

a = -3

. The change in

y

1 - (-2) = 1 - (-2) = 3

, so

b = 3

. The rule is

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

. You can check:

3 - 3 = 0

✓ and

-2 + 3 = 1

✓.

Point

B(5, -3)

is reflected across the y-axis. What are the coordinates of its image

B'

$(-5, -3)$
$(5, 3)$
$(-5, 3)$
$(5, -3)$

Answer: $(-5, -3)$

Reflection across the y-axis uses the rule

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

. The y-coordinate stays the same, and the x-coordinate changes sign. So

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

. A point to the right of the y-axis (positive

x

) flips to the left (negative

x

), but its height above or below the x-axis does not change.

A line segment has endpoints at

(1, 3)

and

(4, 7)

. After the translation

(x, y) \to (x - 2, y + 5)

, what is the length of the image segment?

$5$ units
$6$ units
$7$ units
$8$ units

Answer: $5$ units

First, find the length of the original segment using the distance formula:

d = \sqrt{(4-1)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5

units. Since translations are rigid transformations, they preserve length. The image segment also has length

5

units. You do not need to compute the new coordinates to answer this question—rigidity guarantees the length stays the same.

FAQ

Why do we say translations and reflections are "rigid" transformations?: Rigid transformations preserve the size and shape of figures. The distances between all pairs of points stay the same, and all angle measures stay the same. This means the image figure is congruent to the original. Translations and reflections both have this property, unlike dilations, which change size.
Do reflections across the x-axis and y-axis use different rules?: Yes. Reflection across the x-axis uses $(x, y) \to (x, -y)$ : flip the sign of the y-coordinate. Reflection across the y-axis uses $(x, y) \to (-x, y)$ : flip the sign of the x-coordinate. The axis name tells you which coordinate changes sign—the other one stays the same.
If I translate a figure and then reflect it, do parallel lines stay parallel?: Yes. Each transformation—translation and reflection—preserves parallel lines individually. A translation slides all points the same way, so if two lines were parallel, they still move together and stay parallel. A reflection flips all points consistently across a line, also preserving parallelism. Combining transformations preserves parallelism as well.
How do I know if I applied a coordinate rule correctly?: Substitute the original coordinates of at least two vertices into the rule and plot both the original and image points on a coordinate plane. Check that the movement or flip makes sense (does the figure move the right direction and distance, or is it flipped correctly across the right line?). Also verify that distances or angles look preserved visually—this quick check often catches mistakes in applying the rule.

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