Lesson 7.1 Rigid Motion in a Plane Today, we will learn to… > identify the 3 basic transformations > use transformations in real-life situations

Transformations   The original figure is called the ____________ and the new figure is called the ____________. preimage image

Transformations   Preimage: A , B , C , D Image: A’ , B’ , C’ , D’

R R REFLECTION

R Rotation

R R Translation

Isometries preserve length, angle measures, parallel lines, & distances between points

Theorems 7.1, 7.2, & 7.4 Reflections, translations, and rotations are isometries.

1. Name and describe the transformation. reflection over the y-axis ABC  A’B’C’

2. Name the coordinates of the vertices of the preimage and image. (0,4) (-4,4) (4,4) (-4,0) (4,0)

3. Name and describe the transformation. reflection over x = -1 ABCD  HGFE

4. Is the transformation an isometry? Explain. NO YES NO YES

5. The mapping is a reflection. Which side should have a length of 7? Explain. WX = 7

6. Name the transformation. Find x and y. Reflection x = 40 y = 4

7. Name the transformation. Find x and y. Reflection x = 12 y = 4

reflection a = 73 b = 53 15 c = d = 8 8. Name the transformation. Find a, b, c, and d. reflection a = 73 b = 53 15 c = d = 8

rotation p = 19 q = 3 r = 7.5 9. Name the transformation. Find p, q, and r. rotation p = 19 q = 3 r = 7.5

10. Name the transformation and complete this statement GHI   ____ LKP reflection

reflection translation rotation 11. Name the transformation that maps the unshaded turtle onto the shaded turtle reflection translation rotation

Lesson 7.2 Reflections Today, we will learn to… > identify and use reflections > identify relationships between reflections and line symmetry

Reflection 2 images required

What is the line of reflection? 1. Is this a reflection? What is the line of reflection? YES x = -2

2. Is this a reflection? NO

What is the line of reflection? 3. Is this a reflection? What is the line of reflection? YES y = 1

What is the line of reflection? 4. Is this a reflection? What is the line of reflection? YES y = x

What is the line of reflection? 5. Is this a reflection? What is the line of reflection? YES y = - x

When can I use this in “Real Life?” Finding a minimum distance Telephone Cable - Pole Placement TV cable (Converter Placement) Walking Distances Helps you work smarter not harder

Finding a minimum distance 6. A new telephone pole needs to be Finding a minimum distance 6. A new telephone pole needs to be placed near the road at point C so that the length of telephone cable (AC + CB) is a minimum distance. Two houses are at positions A and B. Where should you locate the telephone pole?

A B C Finding a minimum distance A’ 1) reflect A 2) connect A’ and B 3) mark C

GSP

1 image reflects onto itself Line of Symmetry 1 image reflects onto itself

7. How many lines of symmetry does the figure have? 1 2 3 8 7 4 6 5

8. How many lines of symmetry does the figure have? 2

m  A = 180˚ n can be used to calculate the angle between the mirrors in a kaleidoscope n = the number of lines of symmetry

1 2 8 3 7 4 6 5 180˚ 8 = 22.5˚

180˚ 9 = 20˚ http://kaleidoscopeheaven.org

10. Find the angle needed for the mirrors in this kaleidoscope. 180˚ 4 = 45˚

Project? Example. 1) Identify a reflection in a flag 2) Identify a line of symmetry

Reflection Line of Symmetry

Reflection Line of Symmetry

Lesson 7.3 Rotations Today, we will learn to… students need tracing paper Today, we will learn to… > identify and use rotations

Rotation Direction of Rotation? Center of Rotation? Angle of Rotation?

Clockwise rotation of 60° Angle of Rotation? 60˚ Clockwise rotation of 60° 60˚ Center of Rotation?

Counter-Clockwise rotation of 40°

Theorem 7.3 A reflection followed by a reflection is a rotation. If x˚ is the angle formed by the lines of reflection, then the angle of rotation is 2x°.

A B’ A’ 2x˚ B x˚ A’’ B’’

1. What is the degree of the rotation? 70˚ 140˚

2. What is the degree of the rotation? 55˚ 110˚ 125˚

Figure ABCD Figure A'B'C'D' A’ (2, 2) B ‘ (–1, 4) C ‘ (–1,5) D ‘(1, 5) 3. Use tracing paper to rotate ABCD 90º counterclockwise about the origin. Figure ABCD A (2, –2) B (4, 1) C (5, 1) D (5, –1) Figure A'B'C'D' A’ (2, 2) B ‘ (–1, 4) C ‘ (–1,5) D ‘(1, 5)

Rotational Symmetry A figure has rotational symmetry if it can be mapped onto itself by a rotation of 180˚ or less. I had another dream….

6. Describe the rotations that map the figure onto itself. 8 1 360˚ 8 = 45˚ 2 7 3 6 4 5 45˚ ___ rotational symmetry

Describe the rotations that map the figure onto itself. 360 2 = 180˚ 1 2 180˚ ____ rotational symmetry

Describe the rotations that map the figure onto itself. no ___ rotational symmetry

Describe the rotational symmetry. 360 6 = 1 60˚ 2 6 3 5 4 60˚ rotational symmetry

Which segment represents a 90˚clockwise rotation of AB about P? CD

LF Which segment represents a 90˚counterclockwise rotation of HI about Q? LF

Project? 1) Identify a rotation in a flag 2) Identify rotational symmetry in a flag

60° Rotational symmetry Rotation

Section 7.3 Practice!!! A B C D E J P K M H F G L

Lesson 7.4 Translations and Vectors Today, we will learn to… > identify and use translations

Translation

THEOREM 7.5 One reflection after another in two parallel lines creates a translation. m n R R R

Q ' P ' Q P k m Q '' P '' PP '' is perpendicular to k and m. ______________ PP '' is parallel to QQ'' _______

d Q ' P ' Q P k m Q '' P '' 2d The distance between P and P” is 2d, if d is the distance between the parallel lines.

Name two segments parallel to YY” XX” ZZ”

Find YY” 12 cm XX”= 12 cm 6 cm ZZ”= 12 cm

A translation maps XYZ onto which triangle? X”Y”Z”

Name two lines  to XX” line k line m

(x, y)  (x + 12, y - 20) means to translate the figure… A translation can be described by coordinate notation. (x, y)  (x + a, y + b) describes movement left or right describes movement up or down – + + – (x, y)  (x + 12, y - 20) means to translate the figure… right 12 spaces & down 20 spaces

Use words to describe the translation. 1. (x, y)  (x + 1, y – 9) right 1 space , down 9 spaces 2. (x, y)  (x – 2, y + 7) left 2 spaces, up 7 spaces

(x, y)  (x + 5, y – 3)

Write the coordinate notation described. 3. left 5, down 10 (x , y)  (x – 5, y – 10) 4. up 6 (x , y)  (x, y + 6)

5. Describe the translation with coordinate notation. +3 -2 +3 -2 (x,y)  (x – 2, y + 3)

6. Describe the translation with coordinate notation. -7 -2 -2 -7 -2 -7 -2 -7 (x,y)  (x – 7, y – 2)

(-4, 3)  (-4 + 4, 3 – 5)  (0,-2) (4, -1) (0, 4)  (0 + 4, 4 – 5)  7. A triangle has vertices (-4,3); (0, 4); and (3, 2). Find the coordinates of its image after the translation (x, y)  (x + 4, y – 5) (-4, 3)  (-4 + 4, 3 – 5)  (0,-2) (4, -1) (0, 4)  (0 + 4, 4 – 5)  (3, 2)  (3 + 4, 2 – 5)  (7, -3)

Graphically, it would be… (x, y)  (x + 4, y – 5) (3, 2)  (7, -3) (-4, 3)  (0, -2) (0, 4)  (4, -1)

preimage  image (x, y)  (x + 6, y – 2) 8. Find the image of (-4, 5) (2, 3) (-4, 5)  (-4 + 6, 5 – 2)  ( __, __ ) 9. Find the preimage of (9, 5) (3, 7) ( _ , _ )  ( x + 6, y – 2)  ( 9, 5 ) x + 6 = 9 y – 2 = 5

A vector is a quantity that has both direction and magnitude (size). A vector can be used to describe a translation.

terminal point BA A initial point 3 units up 4 B 2 5 units right 5

Write this in coordinate notation form The vector component form combines the horizontal and vertical components. Write this in coordinate notation form (x, y)  (x + 5, y + 3)

10. What is the component form of the vector used for this translation? 4 A D 2 A' D' -5 B 5 C -2 B' C' -4

XY 11. Name the vector and write its component form. X Y Write this in coordinate form. (x,y)  (x + 5, y – 3)

12). Describe the translation which. maps ABC onto A’B’C’ by 12) Describe the translation which maps ABC onto A’B’C’ by writing the translation in coordinate form and in vector component form. A(3,6); B(1,0); C(4,8); A’(1,2); B’(-1,-4); C’(2,4) (x, y)  (x – 2, y – 4)  – 2, – 4

Project? 1) Identify a translation in a flag

Translation Burundi

Lesson 7.5 Glide Reflections and Compositions students need worksheets and tracing paper

glide reflection R R R R R R Example #1 Example #2 To be a “glide” reflection, the translation must be parallel to the line of reflection.

These are just examples of a translation followed by a reflection. NOT a glide reflection NOT a glide reflection R R R R R R These are just examples of a translation followed by a reflection.

Two or more transformations are combined to create a composition.

1. translation: (x,y) (x, y+2) reflection: in the y-axis A (2, 4) A’ ( , ) A’’ ( , ) 2 6 -2 6

translation: (x,y) (x+2, y-3) 2. reflection: in y = x translation: (x,y) (x+2, y-3) A A’ A” A (-3, -2) A’ ( , ) A’’ ( , ) -2 -3 -6

3. translation: (x,y)  (x-3, y) reflection: in the x-axis and B (5, 1) A’ A B B’ B’’ A” A’’ (-1,- 4) and B’’ ( 2,- 1)

A’’ (-6, 0) and B’’ ( -4,-3) A (0, 4) and B (3, 2). 4. translation: (x,y) (x, y+2) reflection: in y = -x A’ B’ A A (0, 4) and B (3, 2). B A” B” A’’ (-6, 0) and B’’ ( -4,-3)

5. Describe the composition. Reflection: in x-axis (x,y) (x + 6,y + 2) Translation:

6. Describe the composition. Reflection: in y = ½ Rotation: 90˚ clockwise about (1,-3)

How do we get better? Practice

Go to the presentation flag_project.