Transformations Exploring Rigid Motion in a Plane What You Should Learn Why You Should Learn It Goal 1: How to identify the three basic

rigid transformations in a plane Goal 2: How to use transformations to identify patterns and their properties in real life You can use transformations to create visual patterns, such as stencil patterns for the border of a wall Identifying Transformations (flips, slides, turns)

Figures in a plane can be reflected, rotated, or slid to produce new figures. The new figure is the image, and the original figure is the preimage The operation that maps (or moves) the preimage onto the image is called a transformation

Blue: preimage Pink: image 3 Basic Transformations Reflection (flip) Translation (slide)

Rotation (turn) ://standards.nctm.org/document/eexamples/chap6/6.4/index.htm Example 1 Identifying Transformations Identify the transformation shown at the left.

Example 1 Identifying Transformations Translation To obtain ABC, each point of ABC was slid 2 units to the right and 3

units up. Rigid Transformations A transformation is rigid if every image is congruent to its preimage This is an example

of a rigid transformation b/c the pink and blue triangles are congruent Example 2 Identifying Rigid Transformations Which of the following transformations appear to be rigid?

Example 2 Identifying Rigid Transformations Which of the following transformations appear to be rigid? The image is not congruent to the preimage, it is smaller The image is not

congruent to the preimage, it is fatter Definition of Isometry A rigid transformation is called an isometry A transformation in the plane is an isometry if it preserves lengths.

(That is, every segment is congruent to its image) It can be proved that isometries not only preserve lengths, they also preserves angle measures, parallel lines, and betweenness of points Example 3 Preserving Distance and Angle Measure

In the figure at the left, PQR is mapped onto XYZ. The mapping is a rotation. Find the length of XY and the measure of Z Example 3 Preserving Distance and Angle Measure

In the figure at the left, PQR is mapped onto XYZ. The mapping is a rotation. Find the length of XY and the measure of Z B/C a rotation is an isometry, the two triangles are congruent, so XY=PQ=3 and

m Z = m R =35 Note that the statement PQR is mapped onto XYZ implies the correspondence PX, QY, and RZ Example 4 Using Transformations in Real-Life Stenciling a Room

You are using the stencil pattern shown below to create a border in a room. How are the ducks labeled, B, C, D, E, and F related to Duck A? How many times would you use the stencil on a wall that is 11 feet, 2 inches long? Example 4 Using Transformations in Real-Life Stenciling a Room

You are using the stencil pattern shown below to create a border in a room. How are the ducks labeled, B, C, D, E, and F related to Duck A? How many times would you use the stencil on a wall that is 11 feet, 2 inches long? Duck C and E are translations of Duck A Example 4 Using Transformations in Real-Life Stenciling a Room

You are using the stencil pattern shown below to create a border in a room. How are the ducks labeled, B, C, D, E, and F related to Duck A? How many times would you use the stencil on a wall that is 11 feet, 2 inches long? Ducks B,D and F are reflections of Duck A Example 4 Using Transformations in Real-Life Stenciling a Room

You are using the stencil pattern shown below to create a border in a room. How are the ducks labeled, B, C, D, E, and F related to Duck A? How many times would you use the stencil on a wall that is 11 feet, 2 inches long? 112 = 11 x 12 + 2 = 134 inches 134 10 = 13.4, the maximum # of times you can use the stencil pattern (without overlapping) is 13

Example 4 Using Transformations in Real-Life Stenciling a Room You are using the stencil pattern shown below to create a border in a room. How are the ducks labeled, B, C, D, E, and F related to Duck A? How many times would you use the stencil on a wall that is 1 feet, 2 inches long? If you want to spread the patterns out more, you can

use the stencil only 11 times. The patterns then use 110 inches of space. The remaining 24 inches allow the patterns to be 2 inches part, with 2 inches on each end Translations (slides) What You Should Learn Why You Should Learn It How to use properties of translations How to use translations to solve reallife problems

You can use translations to solve real-life problems, such as determining patterns in music A translation (slide) is an isometry The picture is moved 2 feet to the right and The points are moved 3 units to the left and

Prime Notation Prime notation is just a added to a number It shows how to show that a figure has moved The preimage is the blue ABC and the image (after the movement) is ABC

O Using Translations A translation by a vector AA' is a transformation that maps every point P in the plane to a point P', so that the following properties are true.

1. PP' = AA' 2. PP' || AA' or PP' is collinear with AA' Coordinate Notation Coordinate notation is when you write things in terms of x and y coordinates. You will be asked to describe the translation using coordinate notation. When you moved from A to A, how far did your x travel (and the

direction) and how far did your y travel (and the direction). Start at point A and describe how you would get to A: Over two and up three Or (x + 2, y + 3) Formula Summary Coordinate Notation for a translation by (a, b): (x + a, y + b)

Ro tat i on s What You Should Learn Why You Should Learn It How to use properties of rotations How to relate rotations and rotational

symmetry You can use rotations to solve reallife problems, such as determining the symmetry of a clock face Using Rotations A rotation about a point O through x degrees (x) is a transformation that maps every point P in the plane to a point P', so that the following properties are true

1. If P is not Point O, then PO = P'O and m POP' = x 2. If P is point O, then P = P' Examples of Rotation Formula Summary Translations Coordinate Notation for a translation by (a, b):

a, y + b) Vector Notation for a translation by (a, b): Rotations Clockwise (CW): (x, y) (y, -x) 180 (x, y) (-x, 270 (x, y) (-y,

90 (x + Counter-clockwise (CCW): (x, y) (-y, x) 180 (x, y) (-x, -y)y) x)270

(x, y) (y, -x) 90 Rotations What are the coordinates for A? A(3, 1) What are the coordinates for A? A(-1, 3) A

A Example 2 Rotations and Rotational Symmetry Which clock faces have rotational symmetry? For those that do, describe the rotations that map the clock face onto itself.

Example 2 Rotations and Rotational Symmetry Which clock faces have rotational symmetry? For those that do, describe the rotations that map the clock face onto itself. Rotational

symmetry about the center, clockwise or counterclockwise 30,60,90,120,150,180 Moving from one dot to the next is (1/12) of a complete turn or (1/12) of 360 Example 2

Rotations and Rotational Symmetry Which clock faces have rotational symmetry? For those that do, describe the rotations that map the clock face onto itself. Does not have rotational symmetry

Example 2 Rotations and Rotational Symmetry Which clock faces have rotational symmetry? For those that do, describe the rotations that map the clock face onto itself. Rotational symmetry about the

center Clockwise or Counterclockwise 90 or 180 Reflections Reflections What You Should Learn Why You Should Learn It Goal 1: How to use properties of reflections Goal 2: How to relate reflections

and line symmetry You can use reflections to solve real-life problems, such as building a kaleidoscope Using Reflections A reflection in a line L is a transformation that maps every point P in the plane to a point P, so that the following properties are true 1. If P is not on L, then L is the perpendicular bisector of PP

2. If P is on L, then P = P Reflection in the Coordinate Plane Suppose the points in a coordinate plane are reflected in the x-axis. So then every point (x,y) is mapped onto the point (x,-y) P (4,2) is mapped onto P (4,-2) What do you notice

about the x-axis? It is the line of reflection It is the perpendicular bisector of PP Reflection in the line y = x Generalize the results when a point is reflected about the line y = x y=x (1,4) (4,1) (-2,3) (3,-2)

(-4,-3) (-3,-4) Reflection in the line y = x Generalize the results when a point is reflected about the line y = x y= x (x,y) maps to (y,x) Formulas

Translations Coordinate Notation for a translation by (a, b): (x + a, y + b) Vector Notation for a translation by (a, b): Rotations Clockwise (CW): 90 (x, y) (y, -x) 180 (x, y) (-x, -y)

270 (x, y) (-y, x) Counter-clockwise (CCW): 90 (x, y) (-y, x) 180 (x, y) (-x, -y) 270 (x, y) (y, -x) Reflections (x, y) (x, -y) y-axis (x = 0) (x, y) (-x, y) Line y = x (x, y) (y, x)

Line y = -x (x, y) (-y, -x) x-axis (y = 0) Any horizontal line (y = n): (x, y) (x, 2n - y) Any vertical line (x = n): (x, y) (2n x, y) 7 Categories of Frieze Patterns Theorem

If lines L and M are parallel, then a reflection in line L followed by a reflection in line M is a translation. If P'' is the image of P after the two reflections, then PP'' is perpendicular to L and PP'' = 2d, where d is the distance between L and M. Glide Reflections & Compositions

What You Should Learn Why You Should Learn It How to use properties of glide reflections How to use compositions of transformations You can use transformations to solve real-life problems, such as creating computer graphics

Using Glide Reflections A glide reflection is a transformation that consists of a translation by a vector, followed by a reflection in a line that is parallel to the vector Composition When two or more transformations are

combined to produce a single transformation, the result is called a composition of the transformations For instance, a translation can be thought of as composition of two reflections Example 1 Finding the Image of a Glide Reflection Consider

the glide reflection composed O v translation 5,0 of the by the vector , followed by a reflection in the xaxis. Describe the image of ABC Example 1 Finding the Image of a Glide Reflection

O v the 5,0 translation Consider the glide reflection composed of by the vector , followed by a reflection in the x-axis. Describe the image of ABC C'

A' B' The image of ABC is A'B'C' with these vertices: A'(1,1) B' (3,1) C' (3,4)

Theorem The composition of two (or more) isometries is an isometry Because glide reflections are compositions of isometries, this theorem implies that glide reflections are isometries

Example 2 Comparing Compositions Compare the following transformations of ABC. Do they produce congruent images? Concurrent images? Hint: Concurren t means meeting

at the same point Example 2 Comparing Compositions

Compare the following transformations of ABC. Do they produce congruent images? Concurrent images? From Theorem 7.6, you know that both compositions are isometries. Thus the triangles are congruent. The two triangles are not concurrent, the final transformations (pink triangles) do not share the same vertices Does the order in

which you perform two transformations affect the resulting composition? Describe the two transformations in each figure

Does the order in which you perform two transformations affect the resulting composition? Describe the two

transformations in each figure Does the order in which you perform two transformations affect the resulting composition? YES Describe the two transformations in each figure Figure 1: Clockwise rotation of

90 about the origin, followed by a counterclockwise rotation of 90 about the point (2,2) Figure 2: a clockwise rotation of 90 about the point (2,2) , followed by a counterclockwise rotation of 90 about the origin Example 3 Using Translations and Rotations in Tetris

Online Tetri s Frieze Patterns What You Should Learn Why You Should Learn It How to use transformations to classify frieze patterns How to use frieze patterns in real

life You can use frieze patterns to create decorative borders for reallife objects such as fabric, pottery, and buildings Classifying Frieze Patterns A frieze pattern or strip pattern is a pattern that extends infinitely to the left and right in such a way that the pattern can be mapped onto itself by a horizontal translation

Some frieze patterns can be mapped onto themselves by other transformations: A 180 rotation A reflection about a horizontal line A reflection about a vertical line A horizontal glide reflection Example 1 Examples of Frieze Patterns Name the transformation that

results in the frieze pattern Name the transformation that results in the frieze pattern Horizontal Translation Horizontal Translation Or

180 Rotation Horizontal Translation Or Reflection about a vertical line Horizontal Translation Or Horizontal glide reflection

Frieze Patterns in Real-Life 7 Categories of Frieze Patterns Classifying Frieze Patterns Using a Tree Diagram Example 2 Classifying Frieze Patterns What kind of frieze pattern is

represented? Example 2 Classifying Frieze Patterns What kind of frieze pattern is represented? TRHVG It can be mapped onto itself by a translation, a 180 rotation, a reflection about a horizontal or vertical line, or a glide reflection

Example 3Classifying a Frieze Pattern A portion of the frieze pattern on the above building is shown. Classify the frieze pattern. TRHVG