Chapter 12 Transformations: Shapes in Motion

Transcription

1 Chapter 12 Transformations: Shapes in Motion 1

2 Table of Contents Reflections Day Pages 1-10 SWBAT: Graph Reflections in the Coordinate Plane HW: Pages #11-15 Translations Day Pages SWBAT: Graph Translations in the Coordinate Plane HW: Pages #22-23 Rotations Day Pages SWBAT: Graph Rotations in the Coordinate Plane HW: Pages #29-31 Dilations/Symmetry Day Pages SWBAT: Graph Dilations in the Coordinate Plane and Identify Line/Rotational Symmetry HW: Pages #44-48 Compositions Day Pages SWBAT: Graph Compositions in the Coordinate Plane HW: Pages #58-62 Overall Review Day Pages SWBAT: Graph Transformations in the Coordinate Plane TRANSFORMATIONS RULES...Page 67

3 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! 1

4 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. 2

5 Example 1: Example 2: Example 3: Match each of the following 1) Rotation a) 2) Translation b) 3) Line Reflection c) 4) Dilation d) 3

6 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 10 y 10 y A C B x A C B x

7 Reflections in the Coordinate Plane

8 5. Triangle: A(1,2), B(1,4) and C(3,3). Reflection: Origin A ( ) B ( ) C ( ) Rule: r origin (x,y) = (, ) Constructions and Line Reflections. Example 1: ABC is reflected across DE 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 DE as point O. What is true about AO and A O? How do you know this is true? 6

9 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 DE 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. DE 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. 7

10 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. 8

11 Identifying a Reflection Write a rule to describe each transformation Challenge 9

12 Summary Exit Ticket Homework: 10

13 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

14 Graphing Reflections 12

15 Identifying Reflections 13

16 Construct the line of reflection for each pair of figures below Reflect the given image across the line of reflection provided. 14

17 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. 15

18 Translations Day 2 Warm - Up: A translation is a rigid motion. A translation is a direct isometry. It preserves distance and orientation. 16

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20 Part b. What is the magnitude of the vector? 18

21 Practice Problems 1. The vertices of ΔLMN are L(2,2), M(5,3) and N(9,1). Translate ΔLMN using the vector <-2, 6 >. 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). 3. Δ DEF is a translation of ΔABC. Use the diagram to write a rule for the translation of ΔABC to ΔDEF. 19

22 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

23 Challenge Summary Exit Ticket 21

24 Translations Day 2 HW a) Write a motion rule for the translation. b) What is the magnitude of the vector? 4. Write a rule for the translation. 22

25 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

26 Rotations Day 3 Warm - Up A rotation is a rigid motion. 24

27 Rules (x, y) R 90 = (, ) R 180 = (, ) R 270 = (, ) 25

28 Practice (2, 3) (-1, 6) (-7, -4) Given triangle X(2,2), Y(4,3) and Z(3,0). Write the coordinates of its image after a rotation Rule: (x,y) ( ) X ( ) Y ( ) Z ( ) 26

29 8. Given triangle A(-2,-5), B(-1,1) and C(1,-2). Write the coordinates of its image after a rotation 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: 27

30 Identifying a Rotation Write a rule to describe each transformation. a. b. Challenge: 28

31 Rotations Homework: 1. Graph the image of the figure under the transformation given and write the coordinates of the image. a) b) c) 29

32 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 180 o. 30

33 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. 31

34 Dilations/Symmetry Day 4 Warm Up. 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 32

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37 8. What happens when the center of dilation is a vertex of the shape? D Dilate ABC from C using a scale factor of 2 D ( ) C,2 ABC F E A B C 9. What happens when the center of dilation is inside the shape? a) Dilate ABC from G using a scale factor of 3 D ( ) G,3 ABC A G B D C H E F b) Dilate ABC from G using a scale factor of ½ D 1 ( ABC) G, 2 A D G B H E C F 35

38 10. What happens when the center of dilation is outside the shape? Dilate ABC from G using a scale factor of 2 D ( ) G,2 ABC G A B C D E H F 11. What happens when the scale factor is negative? Dilate DEF from H using a scale factor of -½ D 1 H, 2 ( DEF ) G C F A D H E B 12. Work backwards to find the center of dilation, and also determine the scale factor. a) Center (, ) Scale Factor = b) Center (, ) Scale Factor = C' C C 4 A' 4 C' A' A 2 A 2 B B' B' 5 5 B c) Center (, ) Scale Factor = d) Center (, ) Scale Factor = 36

39 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 2 x + 4 B) y = 1 2 x + 8 m 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? m y P x A) B) C) D) y y y m' m' m' y m' P P x P x x P x Why? 37

40 Directed Line Segments If we move the initial point (the center of dilation) then we adjust our relationship to represent that change. D( a, b), k 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). 38

41 Example: Use a compass and straightedge to construct the dilation of LMN with the given center and scale factor 3. 39

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44 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. 42

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46 Homework:

47 4. 5. Graph B(-5,-10), C(-5,0) and D(0,5) and its image after a dilation of -1/

48 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. 46

49 9. Use a compass and straightedge to construct the dilation of LMN with the given center and scale factor 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). 47

50 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 m 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? y m P x

51 Compositions and Rigid Motions Day 5 Warm Up 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. 49

52 Example 1: Graph RS 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 50

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54 Example 3: Example 4: 52

55 Example 5: 53

56 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? 54

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58 Example 9: Example 10: 56

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60 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 r x axis o R 90 (-3,5). b) What single transformation would be the same as the composition in part a? 58

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63 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) 61

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

65 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 What is the length of A B, the image of AB after a dilation with a scale factor of 3? a) 5 10 b) 2 30 c) 6 10 d ) Which transformation is represented in the table below? a) T -1,3 b) T 1,3 c) T -3,1 d) T 1,-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? o o a) T (2,5) b) R 90 c)r 180 d) r x-axis 9. Which of the following composition of transformations maps ABC to A B C? 1) R 180 o o T -2,2 2) T 1,4 o r y-axis 3) T 3,-2 o r x-axis 4) R 90 o o r y=x 63

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67 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. 65

68 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 R 180 o o r x-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. 66

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