Transformations I- Math for Interior design

TRANSFORMATIONS-I(TRANSLATIONS,

ROTATIONS, REFLECTIONS, AND DILATIONS)

Math 103GEOMETRY

Dr. Farhana Shaheen

OBJECTIVES: TRANSFORMATION Demonstrate understanding of: 1. Translations 2. Rotations 3. Reflections 4. Dilations Relate symmetry to appropriate

transformations.

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In geometry, a transformation is a way to change the position of a figure.

Transformation 05/02/2023

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Types of TransformationsTranslations, Rotations, Reflections, and

Dilations

Transformation is a way to change.

Transformation in Real life 05/02/2023

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In some transformations, the figure retains its size and only its position is changed.

Examples of this type of transformation are:translations, rotations, and reflections

In other transformations, such as dilations, the size of the figure will change.

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TRANSLATION 05/02/2023

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TRANSLATIONA translation is a transformation that slides a figure across a plane or through space.

With translation all points of a figure move the same distance and the same direction.

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TRANSLATIONBasically, translation means that a figure has moved.

An easy way to remember what translation means is to remember…

A TRANSLATION IS A CHANGE IN LOCATION.

A translation is usually specified by a direction and a distance.

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TRANSLATIONWhat does a translation look like?

A TRANSLATION IS A CHANGE IN LOCATION.

x yTranslate from x to y

original image

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TRANSLATIONIn the example below triangle A is translated to

become triangle B.

A B

Describe the translation.Triangle A is slide directly to the right.

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TRANSLATIONIn the example below arrow A is translated to

become arrow B.

Describe the translation.Arrow A is slide down and to the right.

A

B

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Example: 1 GUESS THE TRANSFORMATION? How did you get it? 05/02/2023

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REFLECTION

A REFLECTION IS FLIPPED OVER A LINE.

A reflection is a transformation that flips a figure across a line.

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REFLECTION

A REFLECTION IS FLIPPED OVER A LINE.

After a shape is reflected, it looks like a mirror image of itself.

Remember, it is the same, but it is backwards

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REFLECTION

REFLECTIONThe line that a shape is flipped over is called a line of reflection.

A REFLECTION IS FLIPPED OVER A LINE.

Line of reflection

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REFLECTIONDetermine if each set of figures shows a reflection or a translation.

A REFLECTION IS FLIPPED OVER A LINE.

A

BC

A’

B’C’

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REFLECTION over x-axis

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REFLECTION over y-axis(x, y) ------- (-x, y)

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REFLECTION in the line y=x(x, y) ------- (y, x)

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REFLECTIONSometimes, a figure hasreflectional symmetry.

This means that it can be folded along a line of reflection within itself so that the two halves of the figure match exactly, point by point.

Basically, if you can fold a shape in half and it matches up exactly, it has reflectional symmetry.

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REFLECTIONAL SYMMETRYAn easy way to understand reflectional symmetry is to

think about folding.

Do you remember folding a piece of

paper, drawing half of a heart, and then

cutting it out?

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REFLECTIONAL SYMMETRYHow many lines of symmetry does each shape have?

Do you see a pattern?

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REFLECTIONAL SYMMETRYWhich of these flags have reflectional symmetry?

United States of America

Mexico

Canada

England

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ROTATION 05/02/2023

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ROTATIONA rotation is a transformation that turns a figure about (around) a point or a line.Basically, rotation means to spin a shape.The point a figure turns around is called the center of rotation. The center of rotation can be on or outside the shape.

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ROTATIONWhat does a rotation look like?

A ROTATION MEANS TO TURN A FIGURE

center of rotation

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ROTATIONThis is another way rotation looks

A ROTATION MEANS TO TURN A FIGURE

The triangle was rotated around the point.

center of rotation

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ROTATIONIf a shape spins

360, how far does it spin?

All the way around

This is called one full turn.

360

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ROTATIONIf a shape spins 90,

how far does it spin?

One-quarter of the way around(x, y) ------- ( y, -x)

This is called a ¼ turn.

90

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ROTATIONIf a shape spins

180, how far does it spin?

Half of the way around

(x, y) ------(-x, -y)

This is called a ½ turn.

180

Rotating a shape 180 turns a shape upside

down.

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ROTATIONIf a shape spins 270 clockwise, how far

does it spin?

3/4th of the way around

(x, y) ------(-y, x)This is called a 3/4 turn.

270

Rotating a shape 270 05/02/2023

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(Anti-clockwise ROTATION)

ROTATIONDescribe how the triangle A was transformed to

make triangle B

A B

Describe the translation.Triangle A was rotated right 90

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ROTATIONDescribe how the arrow A was transformed to make

arrow B

Arrow A was rotated right 180

A

B

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ROTATIONWhen some shapes are rotated they

create a special situation called rotational symmetry.

to spin a shape the exact same

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ROTATIONAL SYMMETRYA shape has rotational symmetry if, after you rotate less than one full turn, it is the

same as the original shape.Here is an example…

If this shape is rotated 360, is it ever the same before the shape returns to

its original direction?When it is rotated 90 it is the same

as it was in the beginning.So this shape is said to have

rotational symmetry.

90

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ROTATIONAL SYMMETRY

Here is another example…As this shape is rotated 360, is it ever the same before the shape returns to

its original direction?When it is rotated 180 it is the same as it was in the beginning.

So this shape is said to have rotational symmetry.180

A shape has rotational symmetry if, after you rotate less than one full turn, it is the

same as the original shape.

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ROTATIONAL SYMMETRY

Here is another example…As this shape is rotated 360, is it ever the same before the shape returns to

its original direction?No, when it is rotated 180 it is

never the same.So this shape does NOT have

rotational symmetry.

A shape has rotational symmetry if, after you rotate less than one full turn, it is the

same as the original shape.

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ROTATION SYMMETRYDoes this shape have rotational symmetry?

120

Yes, when the shape is rotated

120 it is the same. Since 120 is less than 360, this shape HAS

rotational symmetry

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CONCLUSION: A rotation is the turning of a figure or object

around a fixed point. And a translation is a scenario where every

point in a figure is moved the exact same distance and in the same exact direction, without being rotated, reflected, or resized.

Translations Reflections and Rotations - YouTube

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CONCLUSIONWe just discussed three types of transformations.

See if you can match the action with the appropriate transformation.

FLIPSLIDETURN

REFLECTIONTRANSLATION

ROTATION

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