Chapter 7 Transformations. Chapter Objectives Identify different types of transformations Define...

Chapter 7

Transformations

Chapter Objectives

Identify different types of transformations

Define isometry

Identify reflection

Identify rotations

Identify translations

Describe composition transformations

Lesson 7.1

Rigid Motion in a Plane

Lesson 7.1 Objectives

Identify basic rigid transformations

Define isometry

Definition of Transformation

A transformation is any operation that maps, or moves, an object to another location or orientation.

Transformation Terms

When performing a transformation, the original figure is called the pre-image.

The new figure is called the image.

Many transformations involve labels

The image is named after the pre-image, by adding a prime symbol (apostrophe)

A A’ A’’

We say it as “A prime”

Types of Transformations

Types Reflection Rotation Translation

Characteristics

Orientation

Pictures

Flips object over line of reflection

Turns object using a fixed point as center or rotation

Slides object through a plane

Order in which object is drawn is

reversed

Stays same just tilted

Stays same and stays upright

Definition of Isometry

An isometry is a transformation that preserves length.

Isometry also preserve angle measures, parallel lines, and distances between points.

If you look at the meaning of the two parts of the word, iso- means same, and metry- means meter or measure.

So simply stated, isometry preserves size.

Any transformation that is an isometry is called a Rigid Transformation.

Homework 7.1

1-33, 36-39p399-401

In Class – 9, 13, 27, 33

Due Tomorrow

Lesson 7.2

Reflections

Lesson 7.2 Objectives

Utilize reflections in a plane

Define line symmetry

Derive formulas for specific reflections in the plane

Reflections

A transformation that uses a line like a mirror is called a reflection.

The line that acts like a mirror is called the line of reflection.

When you talk of a reflection, you must include your line of reflection

A reflection in a line m is a transformation that maps every point P in the plane to a point P’, so that the following is true If P is not on line m, then m is the perpendicular bisector of PP’. If P is on line m, then P=P’.

Theorem 7.1:Reflection Theorem

A reflection is an isometry.That means a reflection does not change

the shape or size of an object!

m

Line of SymmetryA figure in the plane has a line of symmetry if the figure can be mapped onto itself by a reflection in a line.What that means is a line can be drawn through an object so that each side reflects onto itself.There can be more than one line of symmetry, in fact a circle has infinitely many around.

Homework 7.2a

1-11, 22-29p407-408

In Class – 7, 23

Due Tomorrow

Reflection Formula

There is a formula to all reflections.

It depends on which type of a line are you reflecting in. vertical horizontal y = x

Vertical:y-axisx = a

( -x + 2a , y)

Horizontal:

x-axisy = a

( x , -y + 2a)

y = x

( y , x)

( x , y)

Homework 7.2b

12-14, 18-21, 50-51p407-410

In Class – 19

Due Tomorrow

Lesson 7.3

Rotations

Lesson 7.3 Objectives

Utilize a rotation in a plane

Define rotational symmetry

Observe any patterns for rotations about the origin

Definitions of Rotations

A rotation is a transformation in which a figure is turned about a fixed point.

The fixed point is called the center of rotation.

The amount that the object is turned is the angle of rotation.

A clockwise rotation will have a negative measurement.

A counterclockwise rotation will have a positive measurement.

Q

clockwiseornegative (-)

Theorem 7.2:Rotation Theorem

A rotation is an isometry.

A

B

A’

B’

P

Rotational Symmetry

A figure in a plane has rotational symmetry if the figure can be mapped onto itself by a rotation of 180° or less. A square has rotational symmetry because it maps onto

itself with a 90° rotation, which is less than 180°. A rectangle has rotational symmetry because it maps onto

itself with a 180° rotation.

Homework 7.3a

1-19p416

In Class – 6, 11, 13

Due Tomorrow

Rotating About the Origin

Rotating about the origin in 90o turns is like reflecting in the line y = x and in an axis at the same time!

So that means to switch the positions of x and y. (x,y) (y,x)

Then the original x-value changes sign, no matter where it is flipped to.

So overall the transformation can be described by (x,y) (-y,x)

Every time you 90o you repeat the process. So going 180o means you do the process twice!

Theorem 7.3:Angle of Rotation Theorem

The angle of rotation is twice as big as the angle of intersection. But the intersection must be the center of rotation. And the angle of intersection must be acute or right only.

P

A

B

A’

B’

m

k

x

2x

Homework 7.3b

25-35, 45-50, 54p417-419

In Class – 25, 35

Due Tomorrow

Quiz WednesdayLessons 7.1-7.4

Lesson 7.4

Translations

and

Vectors

Lesson 7.4

Define a translation

Identify a translation in a plane

Use vectors to describe a translation

Identify vector notation

Translation Definition

A translation is a transformation that maps an object by shifting or sliding the object and all of its parts in a straight light.

A translation must also move the entire object the same distance.

Theorem 7.4:Translation Theorem

A translation is an isometry.

Theorem 7.5:Distance of Translation Theorem

The distance of the translation is twice the distance between the reflecting lines.

P

Q

P’

Q’

P’’

Q’’

x

2x

k m

Coordinate form

Every translation has a horizontal movement and a vertical movement.

A translation can be described in coordinate notation. (x,y) (x+a , y+b)Which tells you to move a units horizontal

and b units vertical.

a units to the right

b units upP

Q

Vectors

Another way to describe a translation is to use a vector.

A vector is a quantity that shows both direction and magnitude, or size. It is represented by an arrow pointing from pre-

image to image. The starting point at the pre-image is called the initial

point. The ending point at the image is called the terminal

point.

Component Form of Vectors

Component form of a vector is a way of combining the individual movements of a vector into a more simple form. <x , y>

Naming a vector is the same as naming a ray. PQ

x units to the right

y units upP

Q

Use of Vectors

Adding/subtracting vectors Add/subtract x values and then add y values

<2 , 6> + <3 , -4> <5 , 2>

Distributive property of vectors Multiply each component by the constant

5<3 , -4> <15 , -20>

Length of vector Pythagorean Theorem

x2 + y2 = lenght2

Direction of vector Inverse tangent

tan-1 (y/x)

Homework 7.4

1-30, 44-47p425-427

In Class – 3,7,17,25,45

Due Tomorrow

Quiz TomorrowLessons 7.1-7.4

Lesson 7.5

Glide Reflections

and

Compositions

Lesson 7.5 Objectives

Identify a glide reflection in a plane

Represent transformations as compositions of simpler transformations

Glide Reflection Definition

A glide reflection is a transformation in which a reflection and a translation are performed one after another.

The translation must be parallel to the line of reflection. As long as this is true, then the order in which the

transformation is performed does not matter!

Compositions of Transformations

When two or more transformations are combined to produce a single transformation, the result is called a composition.So a glide reflection is a composition.

The order of compositions is important!A rotation 90o CCW followed by a reflection

in the y-axis yields a different result when performed in a different order.

Theorem 7.6:Composition Theorem

The composition of two (or more) isometries is an isometry.

Homework 7.5

1-8, 9-21, 23-24, 26-30skip 16, 28p433-435

In Class – 9,13,19

Due Tomorrow

Lesson 7.6

Frieze Patterns

Lesson 7.6 Objectives

Identify a frieze pattern by type

Visualize the different compositions of transformations

Frieze Patterns

A frieze pattern is a pattern that extends to the left or right in such a way that the pattern can be mapped onto itself by a horizontal translation.Also called a border pattern.

Classifying Frieze Patterns

The horizontal translation is the minimum that must exist.However, there are other transformations that can occur. And they can occur more than once.

Type Abbreviation Description

Translation T Horizontal translation left or right

180o Rotation R 180o Rotation CW or CCW

Reflection inHorizontal Line

HReflection either up or down

in a horizontal line

Reflection inVertical Line

V Reflection either left or rightin a vertical line

HorizontalGlide Reflection

GHorizontal translation with

reflection in a horizontal line

Examples

TR TG

TV

THG

TRVGTRHVG

Homework 7.6

2-23p440-441

In Class – 9,13,17,21

Due Tomorrow

Quiz TuesdayLessons – 7.5-7.6