Transformations and Congruence

Grade 8 · Math · Study Guide

Explore how shapes move on the coordinate plane through translations, rotations, reflections, and dilations.

A transformation is a change in the position, size, or orientation of a figure. The original figure is called the pre-image, and the result is called the image.

There are four main types of transformations. A translation slides a figure without changing its size or orientation. A reflection flips a figure over a line of reflection (like the x-axis or y-axis). A rotation turns a figure around a center point by a given angle. A dilation enlarges or shrinks a figure using a scale factor.

Translations, reflections, and rotations are called rigid transformations (or isometries) because they preserve size and shape. When a figure undergoes a rigid transformation, the pre-image and image are congruent — they have the same side lengths and angle measures.

Dilations change the size of a figure but preserve the shape and angle measures. A dilation with a scale factor greater than 1 enlarges the figure, while a scale factor between 0 and 1 shrinks it. After a dilation, the figures are similar (same shape) but not necessarily congruent.

Congruent figures have exactly the same shape and size. Two figures are congruent if one can be mapped onto the other using only rigid transformations.

Key terms

Transformation: A change in position, size, or orientation of a figure
Rigid transformation: A transformation that preserves size and shape (translation, reflection, rotation)
Congruent figures: Figures with the same shape and size; one maps onto the other via rigid transformations
Dilation: A transformation that changes the size of a figure by a scale factor
Translation: A transformation that slides a figure in a given direction without rotating or flipping it
Line of reflection: The line over which a figure is flipped in a reflection

Flashcards

Pre-image: The original figure before a transformation
Image: The resulting figure after a transformation
Translation: Slides a figure; every point moves the same distance in the same direction
Reflection: Flips a figure over a line of reflection
Rotation: Turns a figure around a fixed center point by a given angle
Dilation: Scales a figure larger or smaller using a scale factor
Congruent: Same shape and size; related by rigid transformations
Similar: Same shape but possibly different size; related by dilations plus rigid transformations

Questions

Which transformation flips a figure over a line?
1. Translation
2. Rotation
3. Reflection
4. Dilation
Show answer
Answer: Reflection
A reflection flips a figure over a line, creating a mirror image.
After a rotation, the image is congruent to the pre-image.
Show answer
Answer: True
Rotation is a rigid transformation, so it preserves both shape and size, making pre-image and image congruent.
A transformation that preserves size and shape is called a _____ transformation.
Show answer
Answer: rigid
Rigid transformations (translations, reflections, rotations) preserve distances and angle measures.
If point A is at (3, 4) and is translated by the vector (2, -1), where is A'?
1. (1, 5)
2. (5, 3)
3. (6, -4)
4. (3, 3)
Show answer
Answer: (5, 3)
Add the vector: (3+2, 4+(-1)) = (5, 3).
A dilation with a scale factor of 2 produces a figure congruent to the original.
Show answer
Answer: False
A scale factor of 2 doubles all side lengths. The figures are similar but not congruent since sizes differ.

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