Translation, Dilation, and Tessellations

Set of question-and-answer flashcards reviewing definitions, properties, formulas, and classifications for translation, dilation, tessellations, and related patterns.

What is a translation in geometric transformations?

A "slide" that moves every point of a figure the same distance in the same direction without rotating, reflecting, or resizing it.

Why is a translation classified as an isometry (rigid transformation)?

Because it preserves the size and shape of the figure, making the pre-image and image congruent.

Which key measurements are preserved under any translation?

Distance between points, angle measures, orientation, parallelism, area, and perimeter all stay the same.

How is a translation completely described?

By a direction and a distance, usually written as a translation vector (h, k).

What is the coordinate rule for translating a point (x, y) by a vector (h, k)?

(x, y) → (x + h, y + k).

After translating the point (3, 4) by (2, −1), what are the new coordinates?

(5, 3).

Do translations have any fixed points?

No, unless the translation vector is zero.

What is a dilation in geometric transformations?

A transformation that enlarges or reduces a figure from a fixed center by a specific scale factor k.

Is a dilation an isometry? Why or why not?

No. It changes the size of the figure; the image is similar but not congruent to the pre-image unless k = 1 or −1.

Which geometric property does a dilation always preserve?

Shape (angles are preserved), and parallel lines remain parallel.

How does a scale factor k affect lengths, area, and volume in a dilation?

Lengths multiply by |k|, area by k², and volume (in 3-D) by k³.

What happens to orientation in a dilation with negative scale factor?

The figure is reflected through the center of dilation, reversing orientation.

What is the only fixed point in a nontrivial dilation?

The center of dilation.

Classify the dilation when k > 1, 0 < k < 1, and k = 1.

k > 1: enlargement; 0 < k < 1: reduction; k = 1: congruent (no size change).

What is the coordinate rule for dilating a point (x, y) about the origin by scale factor k?

(x, y) → (k x, k y).

Define a tessellation.

A repeating pattern of one or more shapes that completely covers a plane with no gaps or overlaps.

State the vertex-angle condition required for any tessellation.

The sum of the interior angles of the tiles meeting at a vertex must equal 360°.

Give four key properties of a tessellation.

No gaps, no overlaps, covers the entire plane, and forms a repeating (often translational) pattern.

Which three regular polygons can form regular tessellations?

Equilateral triangles, squares, and regular hexagons.

How many of each regular polygon meet at a vertex in the three regular tessellations?

Triangles: 6; Squares: 4; Hexagons: 3.

What distinguishes a semi-regular (Archimedean) tessellation from a regular tessellation?

It uses two or more types of regular polygons arranged in the same pattern at every vertex.

Describe an Escher-type tessellation.

A tiling made of recognizable, often non-polygonal figures (e.g., birds, fish) created by modifying sides of a tessellating polygon through geometric transformations.

Name the four basic rigid transformations used to generate tessellations.

Translation, rotation, reflection, and glide reflection.

How does a rotation create a tessellation?

A tile is turned around a fixed point (often a vertex) by an angle that divides 360°, allowing copies to fit together repeatedly.

Define a glide reflection.

A transformation combining a translation along a line and a reflection across a line parallel to that translation.

What is a Frieze pattern?

A pattern that repeats infinitely in one direction along a strip, like a decorative border.

How are translation and dilation related to tessellations?

Translation and other transformations act on individual tiles; repeating these transformed tiles across the plane produces a tessellation.

Why are tessellations considered a meeting point of mathematics and art?

They illustrate geometric concepts such as symmetry and transformation while creating visually appealing, often intricate patterns.

What is a translation in geometric transformations?

A "slide" that moves every point of a figure the same distance in the same direction without rotating, reflecting, or resizing it.

Why is a translation classified as an isometry (rigid transformation)?

Because it preserves the size and shape of the figure, making the pre-image and image congruent.

Which key measurements are preserved under any translation?

Distance between points, angle measures, orientation, parallelism, area, and perimeter all stay the same.

How is a translation completely described?

By a direction and a distance, usually written as a translation vector (h, k).

What is the coordinate rule for translating a point (x, y) by a vector (h, k)?

(x, y) → (x + h, y + k).

After translating the point (3, 4) by (2, −1), what are the new coordinates?

(5, 3).

Do translations have any fixed points?

No, unless the translation vector is zero.

What is a dilation in geometric transformations?

A transformation that enlarges or reduces a figure from a fixed center by a specific scale factor k.

Is a dilation an isometry? Why or why not?

No. It changes the size of the figure; the image is similar but not congruent to the pre-image unless k = 1 or −1.

Which geometric property does a dilation always preserve?

Shape (angles are preserved), and parallel lines remain parallel.

How does a scale factor k affect lengths, area, and volume in a dilation?

Lengths multiply by |k|, area by k², and volume (in 3-D) by k³.

What happens to orientation in a dilation with negative scale factor?

The figure is reflected through the center of dilation, reversing orientation.

What is the only fixed point in a nontrivial dilation?

The center of dilation.

Classify the dilation when k > 1, 0 < k < 1, and k = 1.

k > 1: enlargement; 0 < k < 1: reduction; k = 1: congruent (no size change).

What is the coordinate rule for dilating a point (x, y) about the origin by scale factor k?

(x, y) → (k x, k y).

Define a tessellation.

A repeating pattern of one or more shapes that completely covers a plane with no gaps or overlaps.

State the vertex-angle condition required for any tessellation.

The sum of the interior angles of the tiles meeting at a vertex must equal 360°.

Give four key properties of a tessellation.

No gaps, no overlaps, covers the entire plane, and forms a repeating (often translational) pattern.

Which three regular polygons can form regular tessellations?

Equilateral triangles, squares, and regular hexagons.

How many of each regular polygon meet at a vertex in the three regular tessellations?

Triangles: 6; Squares: 4; Hexagons: 3.

What distinguishes a semi-regular (Archimedean) tessellation from a regular tessellation?

It uses two or more types of regular polygons arranged in the same pattern at every vertex.

Describe an Escher-type tessellation.

A tiling made of recognizable, often non-polygonal figures (e.g., birds, fish) created by modifying sides of a tessellating polygon through geometric transformations.

Name the four basic rigid transformations used to generate tessellations.

Translation, rotation, reflection, and glide reflection.

How does a rotation create a tessellation?

A tile is turned around a fixed point (often a vertex) by an angle that divides 360°, allowing copies to fit together repeatedly.

Define a glide reflection.

A transformation combining a translation along a line and a reflection across a line parallel to that translation.

What is a Frieze pattern?

A pattern that repeats infinitely in one direction along a strip, like a decorative border.

How are translation and dilation related to tessellations?

Translation and other transformations act on individual tiles; repeating these transformed tiles across the plane produces a tessellation.

They illustrate geometric concepts such as symmetry and transformation while creating visually appealing, often intricate patterns.