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 ’

Rotation

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  ABC  the y-axis A’B’C’A’B’C’

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

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

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

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 =y = 404

Reflection 7. Name the transformation. Find x and y. x =y =124

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

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

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

11. Name the transformation that maps the unshaded turtle onto the shaded turtle reflectiontranslation 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

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

2. Is this a reflection? NO

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

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

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

GSP

Line of Symmetry 1 image reflects onto itself

7. How many lines of symmetry does the figure have?

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

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

˚ 8 = 22.5˚

180˚ 9 = 20˚

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

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

Reflection Line of Symmetry

Reflection Line of Symmetry

Section 7.2 Practice Sheet !!!

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

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

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

Counter- Clockwise rotation of 40° 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’ A’’ B’’ B 2x˚ x˚

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

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

3. Use tracing paper to rotate ABCD 90º counterclockwise about the origin. B (4, 1) Figure ABCD Figure A ' B ' C ' D ' A (2, –2) A ’ (2, 2)B ‘ (–1, 4) C (5, 1) C ‘ (–1,5) D (5, –1) 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. = 45˚ 360˚ ___ rotational symmetry 45˚

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

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

Describe the rotational symmetry = 60˚ 60˚ rotational symmetry

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

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

Rotation 60° Rotational symmetry

A B C D E J P KM HF G L Section 7.3 Practice!!!

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

Translation

One reflection after another in two parallel lines creates a translation. mn THEOREM 7.5

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

km Q P Q 'Q ' P ' Q '' P '' 2d2d d 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 ” 6 cm 12 cm XX ” = ZZ ” = 12 cm

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

Name two lines  to XX ” line k line m

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 – + + –

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

(x, y)  (x + 5, y – 3)

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

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

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

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)  (3, 2)  (3 + 4, 2 – 5)  (0, 4)  (0 + 4, 4 – 5)  (7, -3) (4, -1) (0,-2)

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

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

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

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

C 10. What is the component form of the vector used for this translation?

11. Name the vector and write its component form. XY 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 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

Project? Two Objects Required One Object Only ReflectionLine of Symmetry Rotation Rotational Symmetry Translation

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

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

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

Two or more transformations are combined to create a composition.

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

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

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

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

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

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

Practice How do we get better?