01 / 21

Describing
Transformations

Learn to identify and describe translations, rotations, reflections, and dilations and how they change the position, orientation, and size of figures.

Pre-Algebra

What Is a Transformation?

A transformation is any operation that moves, flips, turns, or resizes a geometric figure. The original figure is called the pre-image, and the result is the image.

There are four main types of transformations you need to know: translations, reflections, rotations, and dilations.

The Four Types at a Glance

Translation
Slide: every point moves the same distance and direction.
Reflection
Flip: the figure is mirrored across a line of reflection.
Rotation
Turn: every point rotates around a center by a given angle.
Dilation
Resize: the figure enlarges or shrinks from a center point.

Rigid vs. Non-Rigid Transformations

Transformations are grouped by whether they preserve size and shape.

Rigid (Isometry)
Preserves both size and shape. The image is congruent to the pre-image. Translations, reflections, and rotations are rigid.
Non-Rigid
Changes size but preserves shape. The image is similar to the pre-image. Dilations are non-rigid.

Translation: The Slide Rule

A B C A' B' C' slide vector

Reflection: The Mirror Flip

y-axis A B C A' B' C'

Rotation: The Turn

center A B C
\( 90^\circ \) CCW
 A' B' C'

Dilation: Resize from a Center

center O A B C A' B' C'
\( k = 2 \)

Congruent vs. Similar

Congruent (\(\cong\)) Similar (\(\sim\))
Same shape? Yes Yes
Same size? Yes No
Angles preserved? Yes Yes
Side lengths preserved? Yes No (proportional)
Produced by Translation, Reflection, Rotation Dilation (\(\pm\) rigid)

How to Describe a Transformation

1
Identify the type. Did the figure slide, flip, turn, or resize?
2
State the specifics. For translations: the direction and distance. For reflections: the line of reflection. For rotations: the center, angle, and direction. For dilations: the center and scale factor.
3
Write the coordinate rule. Express the mapping as \( (x, y) \to (\ldots,\; \ldots) \).
4
Verify with a point. Pick one vertex from the pre-image, apply the rule, and check it matches the image vertex.

Example 1: Translation

(1, 2) (4, 2) (2, 5) (4, −1) (7, −1) (5, 2)

Example 2: Reflection

x-axis (−3, 1) (−1, 4) (2, 2) (−3, −1) (−1, −4) (2, −2)

Example 3: \( 90^\circ \) Rotation

y x (1, 2) (4, 1) (3, 4) (−2, 1) (−1, 4) (−4, 3)
\( 90^\circ \) CCW

Example 4: Dilation with \( k = \tfrac{1}{2} \)

O (2, 6) (8, 6) (8, 2) (2, 2) (1, 3) (4, 3) (4, 1) (1, 1)
\( k = \tfrac{1}{2} \)

Common Mistakes to Avoid

Confusing Direction
Rotating clockwise when the problem says counterclockwise, or reflecting over the wrong axis.
Flipping Coordinates
Writing \( (y, x) \) instead of \( (-y, x) \) for a \( 90^\circ \) CCW rotation. Watch the negative signs.
Dilation is Not Translation
A dilation changes distances from the center, not a uniform slide. Multiplying by \( k \), not adding.

Summary of Transformation Rules

Transformation Rule Preserves
Translation \((a, b)\) \((x, y) \to (x+a,\; y+b)\) Size + Shape
Reflection (x-axis) \((x, y) \to (x,\; -y)\) Size + Shape
Reflection (y-axis) \((x, y) \to (-x,\; y)\) Size + Shape
Rotation \( 90^\circ \) CCW \((x, y) \to (-y,\; x)\) Size + Shape
Rotation \( 180^\circ \) \((x, y) \to (-x,\; -y)\) Size + Shape
Dilation (factor \(k\)) \((x, y) \to (kx,\; ky)\) Shape only

Question 1 of 5

A triangle has vertices \( A(2, 3) \), \( B(5, 3) \), and \( C(4, 7) \). After a transformation, the image vertices are \( A'(2, -3) \), \( B'(5, -3) \), and \( C'(4, -7) \). Which transformation was applied?

Question 2 of 5

Point \( P(3, -1) \) is rotated \( 90^\circ \) counterclockwise about the origin. What are the coordinates of \( P' \)?

Question 3 of 5

A figure is dilated from the origin with a scale factor of \( k = 3 \). If a vertex of the pre-image is at \( (2, -4) \), where is the corresponding image vertex?

Question 4 of 5

Which of the following transformations does not produce a congruent image?

Question 5 of 5

The rule \( (x, y) \to (x + 4,\; y - 2) \) describes which type of transformation?