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Chapter 12 Transformations: Shapes in Motion

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  • 1

    Chapter 12

    Transformations:

    Shapes in Motion

  • Table of Contents

    Reflections – Day 1 ……………………………………………..…………….. Pages 1 - 10

    SWBAT: Graph Reflections in the Coordinate Plane

    HW: Pages #11 - 15

    Translations – Day 2 ……………….……….…………….……..…………….. Pages 16-21

    SWBAT: Graph Translations in the Coordinate Plane

    HW: Pages #22 - 23

    Rotations – Day 3…………………………………………..………………….. Pages 24-28

    SWBAT: Graph Rotations in the Coordinate Plane

    HW: Pages #29-31

    Dilations/Symmetry – Day 4 …………………………………………….….….. Pages 32-43

    SWBAT: Graph Dilations in the Coordinate Plane and Identify Line/Rotational Symmetry

    HW: Pages #44-48

    Compositions – Day 5……………………..……………………………….…….. Pages 49-57

    SWBAT: Graph Compositions in the Coordinate Plane

    HW: Pages #58-62

    Overall Review – Day 6……………………….………………………..………….. Pages 63-66

    SWBAT: Graph Transformations in the Coordinate Plane

    TRANSFORMATIONS RULES ………….…………………………………………..Page 67

  • 1

    Reflections – Day 1

    A transformation is a function that moves or changes a figure in some

    way to produce a new figure call an image. Another name for the

    original figure is the pre-image.

    We will study the following transformations:

    Reflections

    Translations

    Rotations

    Dilations R Resizing!

  • 2

    Vocabulary to know:

    A Rigid Motion is a transformation that preserves length (distance

    preserving) and angle measure (angle preserving).

    Another name for a rigid motion is an isometry.

    Orientation refers to the arrangement of points relevant to one another,

    after a transformation has occurred.

    A direct isometry preserves distance and orientation. Translations and

    Rotations are direct isometries.

    An opposite isometry preserves distance but does not preserve

    orientation. Line reflections are opposite isometries.

  • 3

    Example 1:

    Example 2:

    Example 3: Match each of the following

    1) Rotation a)

    2) Translation b)

    3) Line Reflection c)

    4) Dilation d)

  • 4

    Line Reflections

    Line reflections are isometries that do not preserve orientation.

    Therefore line reflections are opposite isometries.

    Thus, a reflected image in a mirror appears "backwards."

    A line reflection is a rigid motion.

    ____________________________________________________

    Example: Reflect each image after the reflection described.

    x = -3 y = 5

    A

    BC

    2 4 6 8 10–2–4–6–8–10 x

    2

    4

    6

    8

    10

    –2

    –4

    –6

    –8

    –10

    y

    A

    BC

    2 4 6 8 10–2–4–6–8–10 x

    2

    4

    6

    8

    10

    –2

    –4

    –6

    –8

    –10

    y

  • 5

    Reflections in the Coordinate Plane

    1.

    3. 4.

  • 6

    5. Triangle: A(1,2), B(1,4) and C(3,3) .

    Reflection: Origin

    A’( )

    B’( )

    C’( )

    Rule: rorigin (x,y) = ( , )

    Constructions and Line Reflections.

    Example 1:

    △ 𝐴𝐵𝐶 is reflected across 𝐷𝐸̅̅ ̅̅ and maps onto △ 𝐴′𝐵′𝐶′.

    Use your compass and straightedge to construct the perpendicular bisector of each of the segments connecting 𝐴 to 𝐴′, 𝐵 to

    𝐵′, and 𝐶 to 𝐶′. What do you notice about these perpendicular bisectors?

    Label the point at which 𝐴𝐴′̅̅ ̅̅ ̅ intersects 𝐷𝐸̅̅ ̅̅ as point 𝑂. What is true about 𝐴𝑂 and 𝐴′𝑂? How do you know this is true?

  • 7

    Example 2:

    Construct the segment that represents the line of reflection for quadrilateral 𝐴𝐵𝐶𝐷 and its image 𝐴′𝐵′𝐶′𝐷′.

    What is true about each point on 𝐴𝐵𝐶𝐷 and its corresponding point on 𝐴′𝐵′𝐶′𝐷′?

    You have shown that a line of reflection is the perpendicular bisector of segments connecting corresponding points on a figure and

    its reflected image. You have also constructed a line of reflection between a figure and its reflected image. Now we need to explore

    methods for constructing the reflected image itself. The first few steps are provided for you in this next stage.

    Example 3

    The task at hand is to construct the reflection of △ 𝐴𝐵𝐶 over line 𝐷𝐸. Follow the steps below to get started, then complete the

    construction on your own.

    1. Construct circle 𝐴: center 𝐴, with radius such that the circle crosses 𝐷𝐸̅̅ ̅̅ at two points (labeled 𝐹 and 𝐺).

    2. Construct circle 𝐹: center 𝐹, radius 𝐹𝐴 and circle 𝐺: center 𝐺, radius 𝐺𝐴. Label the [unlabeled] point of intersection

    between circles 𝐹 and 𝐺 as point 𝐴′. This is the reflection of vertex 𝐴 across 𝐷𝐸̅̅ ̅̅ .

    3. Repeat steps 1 and 2 for vertices 𝐵 and 𝐶 to locate 𝐵′ and 𝐶′.

    4. Connect 𝐴′, 𝐵′, and 𝐶′ to construct the reflected triangle.

  • 8

    Things to consider:

    When you found the line of reflection earlier, you did this by constructing perpendicular bisectors of segments joining two

    corresponding vertices. How does the reflection you constructed above relate to your earlier efforts at finding the line of reflection

    itself? Why did the construction above work?

    Example 4

    Now try a slightly more complex figure. Reflect 𝐴𝐵𝐶𝐷 across line 𝐸𝐹.

  • 9

    Identifying a Reflection

    Write a rule to describe each transformation.

    1. 2.

    Challenge

  • 10

    Summary

    Exit Ticket

    Homework:

  • 11

    Homework for Reflections – Day 1

    1. Find the reflection of the point (5,7) under a reflection in the x-axis.

    2. Find the reflection of the point (6,-2) under a reflection in the y-axis.

    3. Find the reflection of the point (-1,-6) under a reflection in the line y =x.

    4.

    5.

  • 12

    Graphing Reflections

  • 13

    Identifying Reflections

  • 14

    Construct the line of reflection for each pair of figures below.

    1.

    2.

    3.

    4. Reflect the given image across the line of

    reflection provided.

  • 15

    5. Draw a triangle 𝐴𝐵𝐶. Draw a line 𝑙 through vertex 𝐶 so that it intersects the triangle at more than just the

    vertex. Construct the reflection across 𝑙.

  • 16

    Translations – Day 2

    Warm - Up:

    1.

    2.

    A translation is a rigid motion. A translation is a direct

    isometry. It preserves distance and orientation.

  • 17

  • 18

    Part b. What is the magnitude of the vector?

  • 19

    Practice Problems

    1. The vertices of ΔLMN are L(2,2), M(5,3) and N(9,1). Translate ΔLMN using the vector .

    2. Graph ΔRST with vertices R(2,2), S(5,2) and T(3,5) and its image under the translation

    (x,y) ( x+ 1, y + 2).

    Triangle 3. Δ DEF is a translation of ΔABC. Use the diagram to write a rule for the translation of ΔABC to ΔDEF.

    DEF is a

  • 20

    4.

    5. a) Write a motion rule for the translation. b) What is the magnitude of the vector?

    6. Describe the composition of transformations shown in the diagram.

    7.

  • 21

    Challenge

    Summary

    Exit Ticket

  • 22

    Translations – Day 2 HW

    1.

    2.

    3. a) Write a motion rule for the translation. b) What is the magnitude of the vector?

    4. Write a rule for the translation.

  • 23

    5. Find the component form of the vector that translates P(-3,6) to P’(0,1).

    6. Describe the composition of transformations shown in the diagram.

    7.

  • 24

    Rotations – Day 3

    Warm - Up

    1.

    2.

    A rotation is a rigid motion.

  • 25

    Rules

    (x, y)

    R90 = ( ___, ____)

    R180 = ( ___, ____)

    R270 = ( ___, ____)

  • 26

    Practice

    3. 4. 5.

    6. Given triangle X(2,2), Y(4,3) and Z(3,0).

    Write the coordinates of its image

    after a rotation 900.

    Rule: (x,y) → ( )

    X’( )

    Y’( )

    Z’( )

    (2, 3) (-1, 6) (-7, -4)

  • 27

    8. Given triangle A(-2,-5), B(-1,1) and C(1,-2).

    Write the coordinates of its image

    after a rotation 2700.

    Rule: (x,y) → ( )

    A’( )

    B’( )

    C’( )

    Finding the center of rotation using constructions

    1. Follow the directions below to locate the center of rotation taking the figure at the top right to its image at the bottom left.

    a. Draw a segment connecting points 𝐴 and 𝐴′.

    b. Using a compass and straightedge, find the perpendicular bisector of this segment.

    c. Draw a segment connecting points 𝐵 and 𝐵′.

    d. Find the perpendicular bisector of this segment.

    e. The point of intersection of the two perpendicular bisectors is the

    center of rotation. Label this point 𝑃.

    2. Find the center of rotation:

  • 28

    Identifying a Rotation Write a rule to describe each transformation.

    a. b.

    Challenge:

  • 29

    Rotations Homework:

    1. Graph the image of the figure under the transformation given and write the coordinates of the image.

    a)

    b)

    c)

  • 30

    2. Write the rule for the transformation.

    3. Write the rule for the transformation.

    4. Graph the image of triangle DEF after a rotation of 180o.

  • 31

    5. Use a compass and a straightedge to find the center of rotation.

    6. On your paper, construct an equilateral triangle. Locate the midpoint of one side using your

    compass. Rotate the triangle 180° around this midpoint. What figure have you formed?

    7. Write the rule for the transformation.

  • 32

    Dilations/Symmetry – Day 4

    Warm – Up

    1. 2.

    A dilation is NOT a rigid motion.

    *It does not preserve distance and is not an

    isometry.

    A dilation preserves:

    1) Angle measure

    2) Parallelism

    3) Colinearity

    4) Midpoint

    5) orientation

  • 33

  • 34

  • 35

    8. What happens when the center of dilation is a vertex of the shape?

    Dilate ABC from C using a scale factor of 2

    ,2( )CD ABC

    9. What happens when the center of dilation is inside the shape?

    a) Dilate ABC from G using a scale factor of 3

    ,3( )GD ABC

    b) Dilate ABC from G using a scale factor of ½

    1,2

    ( )G

    D ABC

    A

    C

    B

    D

    F E

    A

    C

    B

    D

    F

    E

    G

    H

    A

    C

    B

    D

    F

    E

    G

    H

  • 36

    10. What happens when the center of dilation is outside the shape? Dilate ABC from G using a scale factor of 2

    ,2( )GD ABC

    11. What happens when the scale factor is negative? Dilate DEF from H using a scale factor of -½

    1,

    2

    ( )H

    D DEF

    12. Work backwards to find the center of dilation, and also determine the scale factor.

    a) Center (______ , _______) Scale Factor = _____

    b) Center (______ , _______) Scale Factor = _____

    c) Center (______ , _______) Scale Factor = _____

    d) Center (______ , _______) Scale Factor = _____

    A

    C

    B

    D

    F

    E

    G

    H

    A

    C

    B

    D

    F

    E

    GH

    4

    2

    5

    B'

    C'

    A'

    A

    C

    B

    4

    2

    5

    B'

    C'

    A' A

    C

    B

  • 37

    Dilations Involving Lines

    13. Dilate line m about the origin with scale factor 2. What is the equation of the line’s image?

    A) y = 1

    2x + 4 B) y =

    1

    2x + 8

    C) y = 1x + 8 D) None of these

    Why?

    _____________________________________________________________________________

    _____________________________________________________________________________

    14. Dilate line m by a factor 2 about P. Which shows the result of the dilation?

    A) B) C) D)

    Why?

    _____________________________________________________________________________

    _____________________________________________________________________________

    m

    m

    x

    y

    P

    m'

    x

    y

    P

    m'

    x

    y

    P

    m'

    x

    y

    P

    m'

    x

    y

    P

  • 38

    Directed Line Segments

    If we move the initial point (the center of dilation) then we adjust our relationship to represent that change.

    ( , ), ( ), ( )a b kD a k run b k rise

    15. Determine the point P that partitions the directed line segment AB into a ratio of 1:1, where A (-5,2) and B (3,6).

  • 39

    Example: Use a compass and straightedge to construct the dilation of LMN with

    the given center and scale factor 3.

  • 40

  • 41

  • 42

    Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry.

    Tell whether each figure has rotational symmetry. If so, give the angle of rotational symmetry.

  • 43

  • 44

    Homework:

    1.

    2.

    3.

  • 45

    4.

    5. Graph B(-5,-10), C(-5,0) and D(0,5) and its image after a dilation of -1/5.

    6.

  • 46

    7. Find the scale factor of the dilation for each of the following: a)

    b)

    c)

    8. Use a compass and straightedge to construct the dilation of LMN with the given center and scale factor ½ .

  • 47

    9. Use a compass and straightedge to construct the dilation of LMN with the given center and scale factor 2.

    10. Determine the point P that partitions the directed line segment AB into a ratio of 1:3, where A (-2,-5) and B (6,-1).

  • 48

    11. Dilate line m about the origin with scale factor 3. What is the equation of the line’s image?

    A) y = 1x + 2 B) y = 3x + 6

    C) y = 2x + 6 D) None of these

    12. Dilate line m by a factor ½ about P. Which shows the result of the dilation?

    13.

    m

    my

    x

    P

  • 49

    Compositions and Rigid Motions – Day 5

    Warm – Up

    1.

    2.

    Which rotation about point O maps B to D? _______

    A composition of transformations is when two or more

    transformations are combined to find a single transformation.

    The composition of two or more rigid motions is a rigid motion.

  • 50

    Example 1: Graph 𝑅𝑆̅̅̅̅ with endpoints R(1, 3) and S(2, 6) and its image after the composition below.

    ___________________________________________________________________________________

    The symbol for a composition of transformations is an open circle.

    The notation is read as a reflection in the x-axis following a translation of (x+3, y+4). Be careful!!! The process is done in reverse!!

    You may see various notations which represent a composition of transformations:

    could also be indicated by

  • 51

    Example 2:

  • 52

    Example 3:

    Example 4:

  • 53

    Example 5:

  • 54

    Example 6:

    Example 7: Use the figure. The distance between line k and line m is 1.6 centimeters.

    a) The preimage is reflected in line k, then in line m. Describe a single transformation that

    maps the figure on the far left to the figure on the far right.

    b) What is the relationship between PP’ and line k? Explain.

    c) What is the distance between P and P’’?

  • 55

    Example 8:

  • 56

    Example 9:

    Example 10:

  • 57

  • 58

    Compositions of Transformations Homework:

    1. Identify any congruent figures in the coordinate plane.

    2. Describe a congruence transformation that maps Quadrilateral SPQR to ZWXY.

    3. Describe a congruence transformation that maps triangle ABC to triangle EFG.

    4. a) Evaluate the composition of rx –axis o R90 (-3,5). b) What single transformation would be the same as the composition in part a?

  • 59

    5.

    6.

  • 60

    7.

    8.

  • 61

    9.

    10. Determine whether the polygons with the given vertices are congruent. Use transformations to explain your reasoning.

    a)

    b) W(-3,1), X(2,1), Y(4,-4), Z(-5,-4) and C(-1,-3), D(-1,2), E(4,4), F(4,-5)

  • 62

    11. Find the angle of rotation that maps A to A’’.

    12.

  • 63

    Test Prep Questions

    1. Which transformation preserves both distance and angle measure? a) Translation b) horizontal stretch c) vertical stretch d) dilation

    2. Which description is always the composition of two different types of transformations? a) Line reflection b) rotation c) translation d) glide reflection

    3. Which of the following transformations preserve angle measure but not distance? a) Dilation b) translation c) vertical stretch d) rotation

    4. AB has a length of 102 . What is the length of A’B’, the image of AB after a dilation with a

    scale factor of 3?

    a) 105 b) 302 c) 106 d ) 306

    5. Which transformation is represented in the table below? a) T-1,3 b) T1,3 c) T-3,1 d) T1,-3

    Input Output

    (3,5) (4,2)

    (1,0) (2,-3)

    (6,3) (7,0)

    6. Which rigid motion does not preserve orientation? a) line reflection b) rotation c) translation d) dilation

    7. Which transformation is not considered a rigid motion? a) dilation b) rotation c) reflection d) translation

    8. Which transformation is equivalent to a reflection over a point? a) T(2,5) b) R90

    o c)R180o d) rx-axis

    9. Which of the following composition of transformations

    maps ABC to A’B’C’? 1) R180

    oo T-2,2 2) T1,4 o ry-axis 3) T3,-2 o rx-axis 4) R90

    o o ry=x

  • 64

    10.

    11.

    12.

    13.

  • 65

    14. A series of three transformations that, when performed on ABC, finally resulting in A’’’B’’’C’’’. Identify each of the three transformations:

    a) ABC to A’B’C’

    b) A’B’C’ to A’’B’’C’’

    c) A’’B’’C’’ to A’’’B’’’C’’’

    d) Is ABC A’’’B’’’C’’’? Justify your answer.

    e) Identify a single transformation that, when performed on A’’’B’’’C’’’, would result in a

    triangle A’’’B’’’C’’’ such that ABC is not congruent to A’’’B’’’C’’’. Justify your answer.

    15. Use a sequence of rigid motions to show that 1 2. Use coordinate notation. Label the vertices in triangle 2 using A’B’C’ to correspond to the appropriate angles in triangle 1.

  • 66

    16. a) On the accompanying grid, graph and label XYZ with coordinates X(0,2), Y(5,0) and Z(1,3).

    b) On the same grid graph and label X’’Y’’Z’’, the image of XYZ after the composition of transformations R180

    oo rx-axis.

    c) What single transformation maps XYZ to X’’Y’’Z’’?

    17. Given circle A with the equation (x-2)2 + (y + 3)2 = 4. a) Perform the translation of circle A: T(2,-4) and label the translation as circle A’ b) Perform a dilation of circle A’ about its center with a scale factor of 2. Label the image A’’. c) Write the equation of circle A’ and circle A’’.

  • 67