Example 1 Use the coordinate mapping ( x, y ) → ( x + 8, y + 3) to translate ΔSAM to create...
-
Upload
hilda-washington -
Category
Documents
-
view
220 -
download
0
Transcript of Example 1 Use the coordinate mapping ( x, y ) → ( x + 8, y + 3) to translate ΔSAM to create...
Dilations
Objectives:
1. To use dilations to create similar figures
2. 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 factor and a center of dilation.
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.
Dilations
Orig
Ne
in
w Si
al
d
de
e
Sik
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?
This Exploration of Tessellations will guide you through the following:
Exploring Tessellations
Definition ofTessellation
Semi-RegularTessellations
Symmetry inTessellations
RegularTessellations
Create yourown
Tessellation
View artistictessellations
byM.C. Escher
TessellationsAround Us
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?
Are you ready to learn more about Tessellations?
Symmetry inTessellations
Regular Tessellations
Semi-RegularTessellations
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 OctagonHexagonPentagonSquare
Does a Triangle Tessellate?
Regular Tessellations
The shapes fit together without overlapping or leaving gaps, so
the answer is YES.
Does a Square Tessellate?
Regular Tessellations
The shapes fit together without overlapping or leaving gaps, so
the answer is YES.
Does a Pentagon Tessellate?
Regular Tessellations
Gap
The shapes DO NOT fit together because there is a gap. So the
answer is NO.
Does a Hexagon Tessellate?
Regular Tessellations
The shapes fit together without overlapping or leaving gaps, so
the answer is YES.
Hexagon Tessellationin Nature
Does an Octagon Tessellate?
Regular Tessellations
The shapes DO NOT fit together because there are gaps. So the
answer is NO.
Gaps
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.
Are you ready to learn more about Tessellations?
Symmetry inTessellations
Regular Tessellations
Semi-RegularTessellations
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 Octagon &
Square
Square & Triangle Hexagon,
Square & Triangle
Semi-Regular Tessellations
Hexagon & Triangle
Can you think of other ways to arrange these hexagons and triangles?
Semi-Regular Tessellations
Octagon & Square
Many floor tiles have these tessellating patterns.
Look familiar?
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.
Semi-Regular Tessellations
What other semi-regular tessellations can you think of?
Translation
Reflection
Glide Reflection
Symmetry in Tessellations
The four types of Symmetry in Tessellations are:
Rotation
Symmetry in Tessellations
RotationTo 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.
CLICK HERE to view someexamples of rotational symmetry.
Back to Symmetry in Tessellations
TranslationTo 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.
Symmetry in Tessellations
CLICK HERE to view someexamples of translational symmetry.
Back to Symmetry in Tessellations
ReflectionTo 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”.
Symmetry in Tessellations
CLICK HERE to view someexamples of reflection symmetry.
Back to Symmetry in Tessellations
Symmetry in Tessellations
Glide ReflectionA 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.
CLICK HERE to view someexamples of glide reflection symmetry.
Back to Symmetry in Tessellations
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.
Create Your Own Tessellation!
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!
* He was born Maurits Cornelis Escher in 1898, in Leeuwarden, Holland.
M.C. Escher developed the tessellating shape as an art form
*Escher was a graphic artist, who specialized in woodcuts and lithographs.
* 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.
Escher saw tile patterns that gave him ideas for his art work
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 usuallyuse representations of living beings, but usesgeometric patterns,especially symmetric(repeating) patterns.
By “distorting” the basic shapes he changed them into animals, birds, andother figures.The effect can beboth startling and beautiful.