Example 1

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# Example 1 - PowerPoint PPT Presentation

Example 1. Use the coordinate mapping ( x , y ) → ( x + 8, y + 3) to translate Δ SAM to create Δ S’A’M’ . Dilations. Objectives: To use dilations to create similar figures To perform dilations in the coordinate plane using coordinate notation. Dilations.

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## Example 1

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Presentation Transcript
Example 1

Use the coordinate mapping (x, y) → (x + 8, y + 3) to translate ΔSAM to create ΔS’A’M’.

Dilations

Objectives:

• To use dilations to create similar figures
• To perform dilations in the coordinate plane using coordinate notation
Dilations

A dilation is a type of transformation that enlarges or reduces a figure.

The dilation is described by a scale factorand a center of dilation.

Dilations

The scale factor k is the ratio of the length of any side in the image to the length of its corresponding side in the preimage.

Example 2

What happens to any point (x, y) under a dilation centered at the origin with a scale factor of k?

Dilations in the Coordinate Plane

You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor.

Dilations in the Coordinate Plane

You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor.

Enlargement:k > 1.

Dilations in the Coordinate Plane

You can describe a dilation with respect to the origin with the notation (x, y) → (kx, ky), where k is the scale factor.

Reduction:0 < k < 1.

Example 3

Determine if ABCD and A’B’C’D’ are similar figures. If so, identify the scale factor of the dilation that maps ABCD onto A’B’C’D’ as well as the center of dilation.

Is this a reduction or an enlargement?

Example 4

A graph shows PQR with vertices P(2, 4), Q(8, 6), and R(6, 2), and segment ST with endpoints S(5, 10) and T(15, 5). At what coordinate would vertex U be placed to create ΔSUT, a triangle similar to ΔPQR?

Example 5

Figure J’K’L’M’N’ is a dilation of figure JKLMN. Find the coordinates of J’ and M’.

Exploring Tessellations

This Exploration of Tessellations will guide you through the following:

Definition ofTessellation

RegularTessellations

Symmetry inTessellations

TessellationsAround Us

Semi-RegularTessellations

View artistictessellationsbyM.C. Escher

Create yourownTessellation

What is a Tessellation?

A Tessellation is a collection of shapes that fit together to cover a surface without overlapping or leaving gaps.

Tessellations in the World Around Us:

Brick Walls

Floor Tiles

Checkerboards

Honeycombs

Textile Patterns

Art

Can you think of some more?

Regular Tessellations

Semi-RegularTessellations

Symmetry inTessellations

Regular Tessellations

Regular Tessellations consist of only one type of regular polygon.

Do you remember what a regular polygon is?

A regular polygon is a shape in which all of the sides and angles are equal. Some examples are shown here:

Triangle

Square

Pentagon

Hexagon

Octagon

Regular Tessellations

Which regular polygons will fit together without overlapping or leaving gaps to create a Regular Tessellation?

Maybe you can guess which ones will tessellate just by looking at them. But, if you need some help, CLICK on each of the Regular Polygons below to determine which ones will tessellate and which ones won’t:

Triangle

Square

Pentagon

Hexagon

Octagon

Regular Tessellations

Does a Triangle Tessellate?

The shapes fit together without overlapping or leaving gaps, so the answer is YES.

Regular Tessellations

Does a Square Tessellate?

The shapes fit together without overlapping or leaving gaps, so the answer is YES.

Regular Tessellations

Does a Pentagon Tessellate?

Gap

The shapes DO NOT fit together because there is a gap. So the answer is NO.

Regular Tessellations

Does a Hexagon Tessellate?

The shapes fit together without overlapping or leaving gaps, so the answer is YES.

Hexagon Tessellationin Nature

Regular Tessellations

Does an Octagon Tessellate?

Gaps

The shapes DO NOT fit together because there are gaps. So the answer is NO.

Figures that Tessellate
• Find the measure of an angle of a regular polygon using the following formula
• If is a factor of 360, then the n-gon will tessellate

Regular Tessellations

As it turns out, the only regular polygons that tessellate are:

TRIANGLES

SQUARES

HEXAGONS

Summary of Regular Tessellations:

Regular Tessellations consist of only one type of regular polygon. The only three regular polygons that will tessellate are the triangle, square, and hexagon.

Regular Tessellations

Semi-RegularTessellations

Symmetry inTessellations

Semi-Regular Tessellations

Semi-Regular Tessellations consist of more than one type of regular polygon. (Remember that a regular polygon is a shape in which all of the sides and angles are equal.)

How will two or more regular polygons fit together without overlapping or leaving gaps to create a Semi-Regular Tessellation? CLICK on each of the combinations below to see examples of these semi-regular tessellations.

Hexagon & Triangle

Square & Triangle

Hexagon, Square & Triangle

Octagon & Square

Semi-Regular Tessellations

Hexagon & Triangle

Can you think of other ways to arrange these hexagons and triangles?

Semi-Regular Tessellations

Octagon & Square

Look familiar?

Many floor tiles have these tessellating patterns.

Semi-Regular Tessellations

Hexagon, Square, & Triangle

Semi-Regular Tessellations

Summary of Semi-Regular Tessellations:

Semi-Regular Tessellations consist of more than one type of regular polygon. You can arrange any combination of regular polygons to create a semi-regular tessellation, just as long as there are no overlaps and no gaps.

What other semi-regular tessellations can you think of?

Symmetry in Tessellations

The four types of Symmetry in Tessellations are:

Rotation

Translation

Reflection

Glide Reflection

Symmetry in Tessellations

Rotation

To rotate an object means to turn it around. Every rotation has a center and an angle. A tessellation possesses rotational symmetry if it can be rotated through some angle and remain unchanged.

Examples of objects with rotational symmetry include automobile wheels, flowers, and kaleidoscope patterns.

Back to Symmetry in Tessellations

Rotational Symmetry

Back to Rotations

Symmetry in Tessellations

Translation

To translate an object means to move it without rotating or reflecting it. Every translation has a direction and a distance. A tessellation possesses translational symmetry if it can be translated (moved) by some distance and remain unchanged.

A tessellation or pattern with translational symmetry is repeating, like a wallpaper or fabric pattern.

Back to Symmetry in Tessellations

Translational Symmetry

Back to Translations

Symmetry in Tessellations

Reflection

To reflect an object means to produce its mirror image. Every reflection has a mirror line. A tessellation possesses reflection symmetry if it can be mirrored about a line and remain unchanged. A reflection of an “R” is a backwards “R”.

Back to Symmetry in Tessellations

Reflection Symmetry

Back to Reflections

Symmetry in Tessellations

Glide Reflection

A glide reflection combines a reflection with a translation along the direction of the mirror line. Glide reflections are the only type of symmetry that involve more than one step. A tessellation possesses glide reflection symmetry if it can be translated by some distance and mirrored about a line and remain unchanged.

Back to Symmetry in Tessellations

Glide Reflection Symmetry

Back to Glide Reflections

Symmetry in Tessellations

• Summary of Symmetry in Tessellations:
• The four types of Symmetry in Tessellations are:
• Rotation
• Translation
• Reflection
• Glide Reflection
• Each of these types of symmetry can be found in various tessellations in the world around us.

Exploring Tessellations

We have explored tessellations by learning the definition of Tessellations, and discovering them in the world around us.

Exploring Tessellations

We have also learned about Regular Tessellations, Semi-Regular Tessellations, and the four types of Symmetry in Tessellations.

• Now that you’ve learned all about Tessellations, it’s time to create your own.
• You can create your own Tessellation by hand, or by using the computer. It’s your choice!
M.C. Escher developed the tessellating shape as an art form

*Escher was a graphic artist, who specialized in woodcuts and lithographs.

* He was born MauritsCornelisEscher in 1898, in Leeuwarden, Holland.

* His father wanted him to be an architect, but bad grades in school and a love of drawing and design led him to a career in the graphic arts.

His interest began in 1936, when he traveled to Spain and saw the tile patterns used in the Alhambra.

Alhambra Palace

* The Alhambra is a walled city and fortress in Granada, Spain. It was built during the last Islamic Dynasty (1238-1492).

* The palace is lavishly decorated with stone and wood carvings and tile patterns on most of the ceilings, walls, and floors.

The Alhambra Palace is afamous example ofMoorish architecture.It may be the most wellknown Muslim construction.

Islamic art does not usually

use representations of living beings, but uses

geometric patterns,

especially symmetric

(repeating) patterns.

birds,

and

other figures.

The effect can be

both startling and beautiful.

Lets make a simple tessellating shape

Lets make

a simple

Tessellating

shape