Content Standards G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel

Transcription

1 Content Standards G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Mathematical Practices 5 Use appropriate tools strategically. 7 Look for and make use of structure.

2 You identified reflections and verified them as congruence transformations. Draw reflections. Draw reflections in the coordinate plane.

3 line of reflection

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5 Reflect a Figure in a Line Draw the reflected image of quadrilateral WXYZ in line p.

6 Draw the reflected image of quadrilateral ABCD in line n. A. B. C. D.

7 Minimize Distance by Using a Reflection BILLIARDS Suppose that you must bounce the cue ball off side A before it rolls into the pocket at B. Locate the point C along side A that the ball must hit to ensure that it will roll directly toward the pocket.

8 MINIATURE GOLF Omar is playing miniature golf at a local course. Because a wall is blocking his direct shot, he needs to bounce the ball off wall W and hit the hole located at point H. Which of these steps would be needed to determine where on wall W Omar should aim? A. Determine how far the obstructing wall is from the ball. B. Reflect point H over the line formed by wall W. C. Determine the exact length of wall W. D. Find the perpendicular distance from the hole to the wall.

9 Reflect a Figure in a Horizontal or Vertical Line A. Quadrilateral JKLM has vertices J(2, 3), K(3, 2), L (2, 1), and M(0, 1). Graph JKLM and its image over x = 1.

10 Reflect a Figure in a Horizontal or Vertical Line B. Quadrilateral JKLM has vertices J(2, 3), K(3, 2), L (2, 1), and M(0, 1). Graph JKLM and its image over y = 2.

11 A. Quadrilateral ABCD has vertices A(1, 2), B(0, 1), C(1, 2), and D(3, 0). Graph ABCD and its image over x = 2. A. B. C. D.

12 B. Quadrilateral WXYZ has vertices W(2, 4), X(3, 3), Y(2, 0), and Z(0, 2). Graph WXYZ and its image over y = 1. A. B. C. D.

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14 Reflect a Figure in the x- or y-axis A. Graph quadrilateral ABCD with vertices A(1, 1), B(3, 2), C (4, 1), and D(2, 3) and its image reflected in the x-axis.

15 Reflect a Figure in the x- or y-axis B. Graph quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, 1), and D(2, 3) and its reflected image in the y-axis.

16 A. Graph quadrilateral LMNO with vertices L(3, 1), M(5, 2), N(6, 1), and O(4, 3) and its reflected image in the x-axis. Select the correct coordinates for the new quadrilateral L'M'N'O'. A. L'(3, 1), M'(5, 2), N'(6, 1), O'(4, 3) B. L'( 3, 1), M'( 5, 2), N'( 6, 1), O'( 4, 3) C. L'( 3, 1), M'( 5, 2), N'( 6, 1), O'( 4, 3) D. L'(1, 3), M'(2, 5), N'( 1, 6), O'( 3, 4)

17 B. Graph quadrilateral LMNO with vertices L( 1, 0), M (1, 1), N(2, 2), and O(0, 4) and its reflected image under the y-axis. Select the correct coordinates for the point M' in the new quadrilateral L'M'N'O'. A. L'( 1, 0), M'(1, 1), N'(2, 2), O'(0, 4) B. L'(1, 0), M'( 1, 1), N'( 2, 2), O'(0, 4) C. L'(1, 0), M'( 1, 1), N'( 2, 2), O'(0, 4) D. L'(0, 1), M'(1, 1), N'( 2, 2), O'( 4, 0)

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19 Reflect a Figure in the Line y = x Quadrilateral ABCD with vertices A(1, 1), B(3, 2), C(4, 1), and D(2, 3). Graph ABCD and its image under reflection of the line y = x.

20 Quadrilateral EFGH has vertices E( 3, 1), F( 1, 3), G(1, 2), and H( 3, 1). Graph EFGH and its image under reflection of the line y = x. Select the correct coordinates for the point H' in the new quadrilateral E'F'G'H'. A. E'( 3, 1), F'( 1, 3), G'(1, 2), H'( 3, 1) B. E'(3, 1), F'(1, 3), G'( 1, 2), H'(3, 1) C. E'(1, 3), F'(3, 1), G'(2, 1), H'( 1, 3) D. E'( 1, 3), F'( 3, 1), G'( 2, 1), H'(1, 3)

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22 Content Standards G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Mathematical Practices 5 Use appropriate tools strategically. 4 Model with mathematics

23 You found the magnitude and direction of vectors. Draw translations. Draw translations in the coordinate plane.

24 translation vector

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26 Draw a Translation Copy the figure and given translation vector. Then draw the translation of the figure along the translation vector.

27 Which of the following shows the translation of ΔABC along the translation vector? A. B. C. D.

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29 Translations in the Coordinate Plane A. Graph ΔTUV with vertices T( 1, 4), U(6, 2), and V(5, 5) along the vector 3, 2.

30 Translations in the Coordinate Plane B. Graph pentagon PENTA with vertices P(1, 0), E(2, 2), N(4, 1), T(4, 1), and A(2, 2) along the vector 5, 1.

31 A. Graph ΔABC with the vertices A( 3, 2), B(4, 4), C(3, 3) along the vector 1, 3. Choose the correct coordinates for ΔA'B'C'. A. A'( 2, 5), B'(5, 1), C'(4, 6) B. A'( 4, 2), B'(3, 4), C'(2, 3) C. A'(3, 1), B'( 4, 7), C'(1, 0) D. A'( 4, 1), B'(3, 7), C'(2, 0)

32 B. Graph ΔGHJK with the vertices G( 4, 2), H( 4, 3), J(1, 3), K(1, 2) along the vector 2, 2. Choose the correct coordinates for ΔG'H'J'K'. A. G'( 6, 4), H'( 6, 1), J'(1, 1), K'(1, 4) B. G'( 2, 4), H'( 2, 1), J'(3, 1), K'(3, 4) C. G'( 2, 0), H'( 2, 5), J'(3, 5), K'(3, 0) D. G'( 8, 4), H'( 8, 6), J'(2, 6), K'(2, 4)

33 Describing Translations A. ANIMATION The graph shows repeated translations that result in the animation of the raindrop. Describe the translation of the raindrop from position 2 to position 3 in function notation and in words.

34 Describing Translations B. ANIMATION The graph shows repeated translations that result in the animation of the raindrop. Describe the translation of the raindrop from position 3 to position 4 using a translation vector.

35 A. The graph shows repeated translations that result in the animation of the soccer ball. Choose the correct translation of the soccer ball from position 2 to position 3 in function notation. A. (x, y) (x + 3, y + 2) B. (x, y) (x + ( 3), y + ( 2)) C. (x, y) (x + ( 3), y + 2) D. (x, y) (x + 3, y + ( 2))

36 B. The graph shows repeated translations that result in the animation of the soccer ball. Describe the translation of the soccer ball from position 3 to position 4 using a translation vector. A. 2, 2 B. 2, 2 C. 2, 2 D. 2, 2

37 Content Standards G.CO.4 Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Mathematical Practices 2 Reason abstractly and quantitatively. 5 Use appropriate tools strategically.

38 You identified rotations and verified them as congruence transformations. Draw rotations. Draw rotations in the coordinate plane.

39 center of rotation angle of rotation

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41 Draw a Rotation Rotate quadrilateral RSTV 45 counterclockwise about point A.

42 For the diagram, which description best identifies the rotation of triangle ABC around point Q? A. 20 clockwise B. 20 counterclockwise C. 90 clockwise D. 90 counterclockwise

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44 Rotations in the Coordinate Plane Triangle DEF has vertices D( 2, 1), E( 1, 1), and F (1, 1). Graph ΔDEF and its image after a rotation of 115 clockwise about the point G( 4, 2).

45 Triangle ABC has vertices A(1, 2), B(4, 6), and C(1, 6). Draw the image of ΔABC under a rotation of 70 counterclockwise about the point M( 1, 1). A. B. C. D.

46 Hexagon DGJTSR is shown below. What is the image of point T after a 90 counterclockwise rotation about the origin? A (5, 3) B ( 5, 3) C ( 3, 5) D (3, 5) Rotations in the Coordinate Plane

47 Triangle PQR is shown below. What is the image of point Q after a 90 counterclockwise rotation about the origin? A. ( 5, 4) B. ( 5, 4) C. (5, 4) D. (4, 5)

48 Content Standards G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.CO.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another. Mathematical Practices 1 Make sense of problems and persevere in solving them. 4 Model with mathematics.

49 You drew reflections, translations, and rotations. Draw glide reflections and other compositions of isometries in the coordinate plane. Draw compositions of reflections in parallel and intersecting lines.

50 composition of transformations glide reflection

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52 Graph a Glide Reflection Quadrilateral BGTS has vertices B( 3, 4), G( 1, 3), T ( 1, 1), and S( 4, 2). Graph BGTS and its image after a translation along 5, 0 and a reflection in the x-axis.

53 Graph a Glide Reflection

54 Quadrilateral RSTU has vertices R(1, 1), S(4, 2), T (3, 4), and U(1, 3). Graph RSTU and its image after a translation along 4, 1 and a reflection in the x-axis. Which point is located at ( 3, 0)? A. R' B. S' C. T' D. U'

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56 Graph Other Compositions of Isometries ΔTUV has vertices T(2, 1), U(5, 2), and V(3, 4). Graph ΔTUV and its image after a translation along 1, 5 and a rotation 180 about the origin.

57 Graph Other Compositions of Isometries

58 ΔJKL has vertices J(2, 3), K(5, 2), and L(3, 0). Graph ΔTUV and its image after a translation along 3, 1 and a rotation 180 about the origin. What are the new coordinates of L''? A. ( 3, 1) B. ( 6, 1) C. (1, 6) D. ( 1, 6)

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61 Reflect a Figure in Two Lines Copy and reflect figure EFGH in line p and then line q. Then describe a single transformation that maps EFGH onto E''F''G''H''.

62 Copy and reflect figure ABC in line s and then line t. Then describe a single transformation that maps ABC onto A''B''C''. A. ABC is reflected across lines and translated down 2 inches. B. ABC is translated down 2 inches onto A''B''C''. C. ABC is translated down 2 inches and reflected across line t. D. ABC is translated down 4 inches onto A''B''C''.

63 Describe Transformations A. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown.

64 Describe Transformations B. LANDSCAPING Describe the transformations that are combined to create the brick pattern shown.

65 A. What transformation must occur to the brick at point M to further complete the pattern shown here? A. The brick must be rotated 180 counterclockwise about point M. B. The brick must be translated one brick width right of point M. C. The brick must be rotated 90 counterclockwise about point M. D. The brick must be rotated 360 counterclockwise about point M.

66 B. What transformation must occur to the brick at point M to further complete the pattern shown here? A. The two bricks must be translated one brick length to the right of point M. B. The two bricks must be translated one brick length down from point M. C. The two bricks must be rotated 180 counterclockwise about point M. D. The two bricks must be rotated 90 counterclockwise about point M.

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68 Content Standards G.CO.3 Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. Mathematical Practices 4 Model with mathematics. 8 Look for and express regularity in repeated reasoning.

69 You drew reflections and rotations of figures. Identify line and rotational symmetries in two-dimensional figures. Identify line and rotational symmetries in three-dimensional figures.

70 symmetry plane symmetry line symmetry axis symmetry line of symmetry rotational symmetry center of symmetry order of symmetry magnitude of symmetry

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72 Identify Line Symmetry A. KALEIDOSCOPES State whether the object appears to have line symmetry. Write yes or no. If so, draw all lines of symmetry, and state their number.

73 Identify Line Symmetry B. KALEIDOSCOPES State whether the object appears to have line symmetry. Write yes or no. If so, draw all lines of symmetry, and state their number.

74 Identify Line Symmetry C. KALEIDOSCOPES State whether the object appears to have line symmetry. Write yes or no. If so, draw all lines of symmetry, and state their number.

75 A. State whether the figure appears to have line symmetry. Write yes or no. If so, state their number. A. yes; 1 line B. yes; 2 lines C. yes; 3 lines D. no

76 B. State whether the figure appears to have line symmetry. Write yes or no. If so, state their number. A. yes; 1 line B. yes; 2 lines C. yes; 4 lines D. no

77 C. State whether the figure appears to have line symmetry. Write yes or no. If so, state their number. A. yes; 1 line B. yes; 2 lines C. yes; 4 lines D. no

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79 Identify Rotational Symmetry A. State whether the figure has rotational symmetry. Write yes or no. If so, locate the center of symmetry, and state the order and magnitude of symmetry.

80 Identify Rotational Symmetry B. State whether the figure has rotational symmetry. Write yes or no. If so, locate the center of symmetry, and state the order and magnitude of symmetry.

81 Identify Rotational Symmetry C. State whether the figure has rotational symmetry. Write yes or no. If so, locate the center of symmetry, and state the order and magnitude of symmetry.

82 A. State whether the figure has rotational symmetry. If so, state the order and magnitude of symmetry. A. Yes, order 8 and magnitude 45 B. Yes, order 4 and magnitude 90 C. Yes, order 4 and magnitude 180 D. No, the figure does not have rotational symmetry.

83 B. State whether the figure has rotational symmetry. If so, state the order and magnitude of symmetry. A. Yes, order 8 and magnitude 45 B. Yes, order 6 and magnitude 60 C. Yes, order 4 and magnitude 90 D. No, the figure does not have rotational symmetry.

84 C. State whether the figure has rotational symmetry. If so, state the order and magnitude of symmetry. A. Yes, order 3 and magnitude 90 B. Yes, order 4 and magnitude 90 C. Yes, order 2 and magnitude 180 D. No, the figure does not have rotational symmetry.

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86 Three-Dimensional Symmetry A. State whether the figure has plane symmetry, axis symmetry, both, or neither.

87 Three-Dimensional Symmetry B. State whether the figure has plane symmetry, axis symmetry, both, or neither.

88 A. State whether the figure has plane symmetry, axis symmetry, both or neither. A. plane symmetry B. axis symmetry C. both D. neither

89 B. State whether the figure has plane symmetry, axis symmetry, both or neither. A. plane symmetry B. axis symmetry C. both D. neither

90 Content Standards G.CO.2 Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch). G.SRT.1 Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor. Mathematical Practices 1 Make sense of problems and persevere in solving them. 5 Use appropriate tools strategically.

91 You identified dilations and verified them as similarity transformations. Draw dilations. Draw dilations in the coordinate plane.

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93 Draw a Dilation Copy trapezoid PQRS and point C. Then use a ruler to draw the image of trapezoid PQRS under a dilation with center C and scale factor 3.

94 Which diagram shows the dilation image of ΔLMN with center C and? A. B. C. D.

95 Find the Scale Factor of a Dilation PUPPETS To create the illusion of a life-sized image, puppeteers sometimes use a light source to show an enlarged image of a puppet projected on a screen or wall. Suppose that the distance between a light source L and the puppet is 24 inches (LP). To what distance PP' should you place the puppet from the screen to create a 49.5-inch tall shadow (I'M') from a 9-inch puppet?

96 Find the Scale Factor of a Dilation

97 PUPPETS Suppose you have a similar situation with the puppet and light source. The distance between the light source L and the puppet is 30 inches (LP). To what distance should you place the puppet from the screen to create a 54-inch tall shadow (I'M') from a 6-inch puppet? A. 100 inches B. 180 inches C. 220 inches D. 240 inches

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99 Dilations in the Coordinate Plane Trapezoid EFGH has vertices E( 8, 4), F( 4, 8), G(8, 4) and H( 4, 8). Graph the image of EFGH after a dilation centered at the origin with a scale factor of

100 Dilations in the Coordinate Plane

101 Triangle ABC has vertices A( 1, 1), B(2, 2), and C( 1, 2). Find the image of ΔABC after a dilation centered at the origin with a scale factor of 2. Sketch the preimage and the image. A. B. C. D. none of the above