Unit 7. Transformations

Transcription

1 Unit 7 Transformations 1

2 A transformation moves or changes a figure in some way to produce a new figure called an. Another name for the original figure is the. Recall that a translation moves every point of a figure the same distance in the same direction. More specifically, a translation maps, or moves, the points P and Q of a plane figure to the points P (P prime) and Q, so that one of the following statements is true: 1. PP = QQ and PP' QQ ' or 2. PP = QQ and PP ' and QQ ' are collinear. EXAMPLE A: Transform the polygon at right using the ordered pair rule (x, y) (x + 2, y 3). Describe the type of transformation. B A C D An isometry is a transformation that preserves length and angles measure. Isometry is another word for congruence transformations. This can also be called a rigid transformation. EXAMPLE B: ΔABC is translated 1 unit right and 4 units up. Draw the image ΔA B C A(1, 3) A' B(3,0) B' C(4, 2) C' As a general rule in arrow notation form, this transformation can be written as ( xy, ) 2

3 EXAMPLE C: Write a rule for the translation of is an isometry using the distance formula. ABC to A' B' C '. Then verify that the transformation Rule: ( xy, ) AB = BC = AC = A B = B C = A C = EXAMPLE D: Write the rule to translate A' B' C' back to ABC. Rule: ( xy, ) Another way to describe a translation is by using a vector. A vector is a quantity that has both direction and magnitude, or size. A vector is represented in the coordinate plane by an arrow drawn from on point to another. The diagram shows a vector named FG read as vector FG The initial point, or starting point, of the vector is F. The terminal point, or ending point, of the vector is G. The component form of a vector combines the horizontal and vertical components. So, the component form of FG is 5,3. 3

4 EXAMPLE E: Name each vector and write it component form. EXAMPLE F: The vertices of ABC are A(0,3), B(2,4), and C(1,0). Translate ABC using the vector <5,-1>. Use the graph below to graph the image and pre-image. A(0,3) A' B(2,4) B' C(1,0) C' EXAMPLE G: The vertices of LMN are L(2,2), M(5,3), and N(9,1). Translate LMN using the vector <-2,4>. Use the graph below to graph the image and pre-image. L(2,2) L' M(5,3) M' N(9,1) N' 4

5 EXAMPLE H: Graph points T(0,3), U(2, 4) and V(5, -1) to make triangle TUV. a. Use the rule to translate the triangle. Rule: (x, y) (x - 3, y - 1). State the coordinates of T, U and V. T U V b. Using the image of ΔT U V perform an additional translation using the rule (x, y) (x + 3, y - 3). State the new coordinates of ΔT U V. T U V c. Is this new image congruent or similar to the original figure? d. Write a single transformation rule from triangle TUV to triangle T U V. 5

6 Skills Practice 1. Translate the image given the following transformations. Then write the rule and vector component form. a. Right 5 units and up 1 unit. b. left 3 units, down 2 units Rule : Vector: Rule: Vector: 2. Use arrow notation to write a rule that describes the translation shown on the graph. a. b. Rule : Rule: 3. MULTIPLE CHOICE: Write a description of the rule x, y x 7, y 4. a. translation 7 units to the right and 4 units up b. translation 7 units to the left and 4 units down c. translation 7 units to the right and 4 units down d. translation 7 units to the left and 4 units up 6

7 4. Write a general rule that maps the pre-image to the image. 5. Draw the translation of the triangle HOT six units left and one unit down. Label the image H O T.. y H O T x Is the image similar or congruent to ΔHOT? How do you know? Write the component form 6. Find the translation of the quadrilateral WXYZ under the rule (x, y) x 2, y 4. W y Z X Y x 7

8 Use the grid below to answer questions 7 through 9. C E y D -7-8 B A x 7. Find the rule to describe the translation from point A to point B. 8. Find the rule to describe the translation from point C to point D. 9. Find the rule to describe the translation from point E to point A. 10. Quadrilateral ABCD is plotted on the grid below. Draw the translation of polygon ABCD eight units to the left and seven units down. Label the image A B C D. 8

9 11. Quadrilateral PQRS is plotted below. Preform the indicated transformations to form P Q R S. Transform quadrilateral PQRS to quadrilateral P Q R S : 3 units to the left and 4 units down Then Transform quadrilateral P Q R S to quadrilateral P Q R S : x, y x 2, y y S P Q Write the single transformation from PQRS to P Q R S. R x 12. The coordinates of WXYZ are W(-6, -1), X(-5, -3), Y(-2, 3), and Z(-3, 1). After a translation 8 units right and 3 units down, the coordinates of the image are: W X Y Z 13. Which of the following figures is a translation of the shaded figure? List all the figures. 14. A' B' C' is the image of ABC after a translation 7 units left and 3 units up. Graph ABC before the translation. 9

10 15. The coordinates of DEFG are D(2, -1), E(5, -2), F(4, -5), and G(1,-4). a. Describe in coordinate mapping (arrow) notation a translation that will move vertex E to the origin. ( xy, ) b. Write the coordinates of the image using the transformation described in part a. 16. Given the graphs below, write the translations that map the pre-image to the image. a. b. Reflections A reflection is a transformation that uses a line like a mirror to reflect an image. The mirror line is called the line of reflection. A reflection in a line m maps every point P in the plane to a point P, so that for each point one of the following properties is true: If P is not on m, then m is the perpendicular bisector of PP ', or If P is on m, then P=P 10

11 EXAMPLE A: The pre-image below is graphed. Graph the image by reflecting the pre-image over the x- axis. a. What are the coordinates of the image? A(1, 3) A' B(3,0) B' C(4, 2) C' b. In general, what is the rule for reflecting over the x-axis? ( xy, ) EXAMPLE B: The pre-image below is graphed. Graph the image by reflecting the pre-image over the y- axis. a. What are the coordinates of the image? A(1, 3) A' B(3,0) B' C(4, 2) C' b. In general, what is the rule for reflecting over the y-axis? ( xy, ) 11

12 EXAMPLE C: Draw ΔJKL which has coordinates J (0,2), K (3,4), and L (5,1). Then preform the following transformations. a. Reflect JKL over the x-axis to form J' K' L'. J K L b. Reflect J' K' L' over the y-axis to form J'' K'' L''. J K L c. Describe a different combination of two reflections that would move ΔJKL to ΔJ K L. d. Are reflections isometries? EXAMPLE D: Draw the triangle ABC with coordinates A(0,1), B(3,4), and C(5,1). a. Reflect the triangle over the x = -1 line. Write the coordinates of the image. 12

13 EXAMPLE E: Draw the triangle ABC with coordinates A(0,1), B(3,4), and C(5,1). a. Reflect the triangle over the y = -2 line. Write the coordinates of the image. EXAMPLE F: Draw the line of reflection on the graph below. Be sure to label it. EXAMPLE G: Draw the line of reflection on the graph below. Then write the equation of the line of reflection. E D 4 y x D' F F' E'

14 EXAMPLE H: Reflect the pre-image below over the x-axis to create quadrilateral C D E F. Then translate quadrilateral C D E F 3 units left and down 2 units to create quadrilateral C D E F. EXAMPLE I: Describe how you would transform Figure 2 to Figure 2 using one reflection and one translation. EXAMPLE J: Below is a reflection over the y = x line. Notice that the line of reflection s slope is 1. Since we know the line of reflection is the perpendicular bisector of each segment formed between the image and pre-image, then the slope of AA, BB, and CC will be -1 (opposite reciprocals). a. What do you notice about the coordinates of the pre-image and image? b. In general, what is the rule for reflecting over the y = x line? ( xy, ) c. Based off this example, what do you think the rule is for reflecting over the y = -x line? ( xy, ) 14

15 Skills Practice 1. Draw the reflection of the triangle HOT over the x-axis. Label the image H O T. y H O T x Is the image similar or congruent? How do you know? 2. Reflect the quadrilateral below over the x = -2 line. W y Z X Y x 3. Triangle PQR is reflected over the y-axis to create Triangle P Q R. Complete the table values. PQR P' Q' R' P(-3,2) P Q(-3,6) Q R(-7,7) R 15

16 4. Draw triangle XYZ and then preform the indicated transformations. X (2,1) Y ( 6,1) Z (4,4) a. Translate 2 units to the left to create triangle X Y Z b. Then reflect triangle X Y Z over the x-axis to create triangle X Y Z. 5. Describe the transformation below. a. b. c. 6. The point of the heart (H) has a coordinate of ( 5, 7) as shown below. The heart is reflected over the y-axis and then reflected over the x-axis. After both reflections, what are the coordinates of the point H? 16

17 7. MULTIPLE CHOICE: Use the figure below to answer the question. Which graph shows the reflection over the x-axis of the pre-image above? 8. MULTIPLE CHOICE: Polygon STUVW is shown below. After polygon STUVW is reflected across the y-axis, what are the coordinates of S, the image of point S after the transformation? A. ( 5, 2) B. ( 5, 2) C. (5, 2) D. (5, 2) 17

18 9. Which graph below shows the figure reflected over the y-axis? 10. Reflect the following pre-image over the y = -x line. 18

19 Rotations Recall that a rotation is a transformation in which a figure is turned about a fixed point called the center of rotation. Rays drawn from the center of rotation to a point and its image form the angle of rotation. A rotation about a point P through an angle of x maps every point Q in the plane to a point Q so that one of the following properties is true: If Q is not the center of rotation P, then QP = Q P and m QPQ ' x, or If Q is the center of rotation P, then the image of Q is Q. A 40 counterclockwise rotation is shown at the right. Rotations can be counterclockwise and clockwise. To rotate a figure: 1. Draw a segment from one vertex to the point of rotation. 2. Use the segment drawn, to draw a ray from your point of rotation to form the angle of rotation. 3. Draw a segment on the ray drawn that is congruent to the segment of the pre-image to the center of rotation. 4. Repeat with other vertices. Example: Draw a 120 rotation of ABC about P. 19

20 1. Triangle ABC is labeled on your graph below. a) Rotate Triangle ABC, 90 o counterclockwise about the origin. Label the triangle A B C. b) Rotate Triangle ABC, 180 o counterclockwise about the origin. Label the triangle A B C. c) Rotate Triangle ABC, 270 o counterclockwise about the origin. Label the triangle A B C. d) Rotate Δ ABC, 360 o counterclockwise about the origin. Label the triangle A B C. 2. Organize your results from Part A in the table. Starting Point 90 Rotation CC 180 Rotation CC 270 Rotation CC 360 Rotation CC A (1, 4) B (5, 2) C (2, 0) 20

21 3. Complete each rule for finding the image of any point (x, y) under the given rotation. a) 90 CCW rotation about the origin: (x, y) (, ) b) 180 CCW rotation about the origin: (x, y) (, ) c) 270 CCW rotation about the origin: (x, y) (, ) d) 360 CCW rotation about the origin: (x, y) (, ) 4. What are the coordinates of (3, - 2) under a 90 counterclockwise rotation about the origin? 5. What are the coordinates of (- 5, 4) under a 180 counterclockwise rotation about the origin? 6. What are the coordinates of ( 3, 2) under a 90 clockwise rotation about the origin? Note: If a direction of rotation is NOT stated, assume the direction is counterclockwise. 7. Draw the final image created by rotating triangle RST 90 counterclockwise about the origin and then reflecting the image in the x-axis. What quadrant is the final image located in? 21

22 8. Draw the final image created by reflecting triangle RST in the x-axis and then rotating the image 90 counterclockwise about the origin. What quadrant is the final image located in? 9. Are the final images in questions 7 and 8 same? Why or why not? 10. Graph Triangle RST with vertices R (2, 3), S (5, 4), and T (4, 8). Use the appropriate rule to rotate the figure 90 counterclockwise about the origin. Write the coordinates of the image. 22

23 11. On the graph, draw the image of quadrilateral ABCD after a counterclockwise rotation of 180 o about the origin. Label the image A B C D. Skills Practice 1. MULTIPLE CHOICE: Point A (3, 6) is rotated 270 counterclockwise about the origin, what is the coordinate of A? Circle the best answer. A. (-6, 3) B. (6, -3) C. (3, 6) D. (-3, -6) 2. Draw the final image after rotating the polygon 90 counterclockwise about the origin. 23

24 3. Determine the transformation that produce the following images. a. b. F' H' D y D' H F 1 x y D H' F' H D' F x 4. Describe how you could move shape 1 to exactly match shape 1 by using a series of transformations. 5. MULTIPLE CHOICE: Suppose quadrilateral QRST is rotated 180 about the origin. In which quadrant y is Q? A. Quadrant I B. Quadrant II C. Quadrant III D. Quadrant IV T Q S R x 24

25 6. Review: Reflect each quadrilateral by the given ordered pair rule. Identify the line of reflection. a. ( x, y) ( x, y) b. ( x, y) ( x, y) c. ( x, y) ( y, x) For Exercises 7-9, transform each quadrilateral by the given ordered pair rule. Identify either the line of reflection or the center of rotation. 10. Describe the type of transformation. Then find the ordered pair rule that transformed triangle PQR to triangle P Q R. ( xy, ) ( xy, ) ( xy, ) 25

26 ( xy, ) ( xy, ) ( xy, ) Match the composition of transformations with the ordered pair rule. 11. ( x, y) ( x h, y k) a. 90 clockwise rotation about the origin. 12. ( x, y) ( x, y) b. reflection across the x- axis 13. ( x, y) ( y, x) c. 90 counterclockwise rotation about the origin 14. ( x, y) ( x, y) d. reflection across the y-axis 15. ( x, y) ( x, y) e. translation by the vector <h, k> 16. ( x, y) ( y, x) f. reflection across the line y = x 17. ( x, y) ( y, x) g. reflection across the line y = -x 18. ( x, y) ( y, x) h. 180 rotation about the origin Dilations A dilation is a transformation in which the original figure and its image are similar. A dilation with center C and scale factor k maps every point P in a figure to point P so that one of the following statements is true: If P is not the center point C, then the image point P lies on CP. The scale factor k is a positive CP' number such that k and k 1, or CP If P is the center point C, then P = P The dilation is a if 0 k 1 and it is an if k > 1. 26

27 EXAMPLE A: Find the scale factor of the dilation. Then tell whether the dilation is a reduction or enlargement. i. ii. EXAMPLE B: Do dilated figures create congruent figures? Explain. Part 1: Dilating from a point that is the origin Learning Task: Dilations in the Coordinate Plane 1. Dilate the following rectangle by a scale factor of 1 2 centered at the origin. Write the coordinates of A, B, C and D. Is the image a reduction or enlargement? A y B x D C 27

28 Part 2: Dilating from a point that is NOT the origin 2. In the example below, we are using the point (1,2) as the center of the dilation. ABC has coordinates A(2, 3), B(4,3) and C(4,6). Find the coordinates dilated with a scale factor of 3. Is the image a reduction or enlargement? y A C B x 3. In the example below, we are using the point (-2,-3) as the center of the dilation. ABC has coordinates A(1, 3), B(4,3) and C(4,6). Find the coordinates dilated with a scale factor of 1. Is the 3 image a reduction or enlargement? y A C B x 28

29 4. In the example below, we are using the point (-2,1) as the center of the dilation. Trapezoid ABCD has coordinates A(-2,1), B(2,1), C(1,-2) and D(-1,-2). Find the coordinates dilated with a scale factor of 2. Is the image a reduction or enlargement? 5 y 4 3 A B x D C 29

30 Part 3: Finding the Scale Factor and Center of Dilation 5. In the figure below, rectangle A B C D is a dilation of rectangle ABCD. a. Determine the center of dilation. Draw lines through the corresponding vertices until you find the point where all of the lines meet. b. Determine the scale factor of the dilation. State if the dilation is an enlargement or reduction D A 1 y D' A' In the figure below, rectangle A B C D is a dilation of rectangle ABCD. C' B' C B x a. Determine the center of dilation. Draw lines through the vertices until you find the point where all of the lines meet. (Note: Use the slope between each corresponding vertex to find points on the line.) b. Determine the scale factor of the dilation. State if the dilation is an enlargement or reduction. D' A' y D A C' B' C B x 30

31 Skills Practice: Dilations 1. Find the coordinates of the vertices of the figure after a dilation of 1 k centered at the origin 2 and graph the image. A (-2, 0) A (, ) B (-2, -6) B (, ) A C(-4, -2) C (, ) C B Series 1 2. Find the coordinates of the vertices of the figure after a dilation of k 2 centered at the origin and graph the image. A (3, 4) A (, ) B (4, 2) B (, ) A C (1, 1) C (, ) C B 31

32 3. Find the coordinates of the vertices of the figure after a dilation of 1 k centered at the point (3, 2 1) and graph the image. A B C 4. Find the coordinates of the vertices of the figure after a dilation of k = 3 centered at the point (-2, 1) and graph the image. L M O P y N x 32

33 5. Find the scale factor and center of dilation for the figure below. B A B' C' C A' D' D 6. Find the scale factor and center of dilation for the figure below. C' C A' B -3-4 B' y A x 33

34 7. MULTIPLE CHOICE: Which transformation of (x, y) is a dilation? A. (3x, y) B. (-x, 3y) C. (3x, 3y) D. ( x + 3, y + 3) 8. A dilation maps A(5,1) to A (2,1) and B(7,4) and B (6,7). Find the scale factor of the dilation. 9. If the scale factor is 5 2 would the image be a reduction or enlargement? Explain. 10. Find the scale factor. Identify if the dilation is an enlargement or reduction. 11. Describe the series of transformations preformed on the polygon ABCD onto polygon A B C D. 34

35 Compositions of Transformations A translation followed by a reflection can be preformed one after the other to produce a glide reflection. A translation can be called the glide. A glide reflection is a transformation in which every point P is mapped to a point P by the following steps: 1. First, a translation maps P to P 2. Then, a reflection in a line k parallel to the direction of the translation maps P to P EXAMPLE A: The vertices of ABC are A(3, 2), B(6, 3), and C(7, 1). Find the image of ABC after the glide reflection. Translation: ( x, y) ( x 12, y) Reflection: over the x- axis When two or more transformations are combined to form a single transformation, the result is a composition of transformations. A glide reflection is an example of a composition of transformations. Write a single translation that is equivalent to the composition of the glide reflection above: Does a glide reflection result in an isometry? Let s look at an example of a composition of two translations. 35

36 EXAMPLE B: Triangle ABC with vertices A( 1, 0), B(4, 0), and C(2, 6) is first translated by the rule (x, y) (x 6, y 5), and then its image, A B C, is translated by the rule (x, y) (x + 12, y + 3) to get A B C. a. What single translation is equivalent to the composition of these two translations? b. What single translation brings the second image, A B C, back to the position of the original triangle, ABC? 36

37 EXAMPLE C: Given PQR with vertices: P( 5, 3), Q( 3, 6), R( 6, 5). Reflect PQR across the y-axis to create P Q R. Reflect P Q R across the line y = 1 to create P Q R. a. What is the transformation rule, (x, y) (?,?), that transforms PQR to P Q R? b. What are the coordinates of the vertices of P Q R? c. What are the coordinates of the vertices of P Q R? d. What is the transformation rule, (x, y) (?,?), that transforms P Q R onto P Q R? e. What is the single transformation rule, (x, y) (?,?), that takes PQR onto P Q R? 37

38 EXAMPLE D: Given DEF with vertices: D( 4, 2), E( 1, 0), F(0, 6). Reflect DEF across the line y = x to create D E F. Rotate DEF 90 clockwise about the origin to create D E F. Reflect D E F across the x-axis to create D E F. a. What is the transformation rule, (x, y) (?,?), that transforms DEF to D E F? b. What are the coordinates of the vertices of D E F? c. What are the coordinates of the vertices of D E F? d. What is the transformation rule, (x, y) (?,?), that transforms DEF to D E F? e. What are the coordinates of the vertices of D E F? f. What is the single transformation rule that takes D E F onto D E F? g. What is the single transformation rule that takes DEF onto D E F? 38

39 Compositions of two reflections result in either a translation or a rotation. EXAMPLE E: Draw quadrilateral ABCD with A(1, -1), B(5, -1), C(5, -3), and D(3, -3). Reflect the quadrilateral over the y-axis to create quadrilateral A B C D. Then reflect quadrilateral A B C D over the x-axis to create quadrilateral A B C D. What single transformation can transform quadrilateral ABCD onto quadrilateral A B C D? EXAMPLE F: Draw quadrilateral ABCD with A(1, -1), B(5, -1), C(5, -3), and D(3, -3). Reflect the quadrilateral over the y-axis to create quadrilateral A B C D. Then reflect quadrilateral A B C D over the x = -6 line to create quadrilateral A B C D. What single transformation can transform quadrilateral ABCD onto quadrilateral A B C D? 39

40 EXAMPLE G: Draw any triangle CQR. Translate your triangle by the transformation rule, (x, y) (x 4, y + 1). Now, reflect this image across the x-axis. What is the single transformation rule that takes your original triangle onto the final image? EXAMPLE H: The vertices of ABC are A(-4, 1), B(-2,2), and C(-2,1). Translate the figure using the rule ( x, y) ( x 5, y 1) to create A' B' C'. Then preform a dilation centered at the origin with a scale factor of 2 to create A'' B'' C'' 40

41 Skills Practice 1. Describe the composition of transformations that result in figure 1 moving onto figure The vertices of FGH are F(-2,-2), G(-2,4), and H(-4,-4). Graph the image of the triangle after a composition of the transformations in the order they are listed. Dilation: centered at the origin with a scale factor of 1 2 Reflection: over the y-axis 41

42 3. The vertices of FGH are F(-2,-2), G(-2,4), and H(-4,-4). Graph the image of the triangle after a composition of the transformations in the order they are listed. Rotation: 90 CCW about the origin Dilation: centered at the origin with a scale factor of Given ABC with vertices: A(-6, -1), B(-4, -3), C(-3, 0) a. Reflect ABC across the x- axis to create A' B' C '. b. Reflect A' B' C' across the line x = -2 to create the image A'' B'' C''. 42

43 c. What single transformation transforms ABC onto A'' B'' C''. 5. Given ABC with vertices: A(-2,3), B(2,2), C(4,7) a. Reflect ABC across the line y = 3 to create A' B' C '. b. Reflect A' B' C' across the line y = -1 to create A'' B'' C''. c. What single transformation transforms ABC onto A'' B'' C''. 6. Given ABC with vertices: A(-1,3), B(3,2), C(5,6) a. Reflect ABC over the x- axis to create A' B' C '. 43

44 b. Translate A' B' C' by the transformation rule ( x, y) ( x 5, y 5) to create A'' B'' C''. c. What single transformation transforms ABC onto A'' B'' C''. 7. Given ABC with vertices: A(-8,2), B(-4,-2), C(-3,3) d. Reflect ABC over the y- axis to create A' B' C '. e. Rotate A' B' C' 90 clockwise about the origin to create A'' B'' C''. f. What single transformation transforms ABC onto A'' B'' C''. 8. MULTIPLE CHOICE: Which transformation will result in an image which is similar, but not congruent, to the pre-image? A. Dilation B. Glide reflection C. Rotation D. Translation 9. MULTIPLE CHOICE: A figure is located entirely in the third quadrant. If it is reflected over the y-axis, what will the signs of the coordinates be? A. Both positive B. Both negative C. x is positive, y is negative D. y is positive, x is negative 10. MULTIPLE CHOICE: QRS is translated by the following vector <-4, 2>. The vertices are as follows: Q (1,3), R(5,1), and S(3,5). Which ordered pair is a vertex of the translated image? 44

45 A. (-1,3) B. (1, -3) C. (1,3) D. (3,1) 11. MULTIPLE CHOICE: JKL has vertices J(2,4), K(3,1), and L(3,3). A translation maps the point J to J (3,3). What are the coordinates of K? A. (-3,1) B. (2,2) C. (3,2) D. (4,0) 12. MULTIPLE CHOICE: Which of the following transformations has the same result as a rotation of 90 clockwise? A. Dilation of a scale factor of 9 B. Rotations of 270 counterclockwise C. Reflection about a horizontal line D. Translation down and to the right 13. MULTIPLE CHOICE: The vertex of a figure is located at (2,4). The figure is rotated and the image of the vertex is located at (-4,-2). Which of these describes the transformation? A. 180 rotation B. 90 counterclockwise rotation C. 270 counterclockwise rotation D. Reflection across the y = - x 14. MULTIPLE CHOICE: A triangle is dilated by a scale factor of 3 with the center of dilation at the origin. Which statement is true? A. The area of the image is nine times the area of the original triangle. B. The perimeter of the image is nine times the perimeter of the original triangle. C. The slope of any side of the image is three times the slope of the corresponding side of the original triangle. D. The measure of each angle in the image is three times the measure of the corresponding angle of the original triangle. 45

46 15. MULTIPLE CHOICE: In the diagram below, a square is graphed in the coordinate plane. A reflection over which line does not carry the square onto itself? A. x = 5 B. y = 2 C. y = x D. y = - x The diagonals of the regular hexagon shown form six equilateral triangles. Use the diagram to complete the statement. a. A clockwise rotation of 60 about P maps R onto b. A counterclockwise rotation of 60 about maps R onto Q. c. A clockwise rotation of 120 about P maps R onto d. A counterclockwise rotation of 180 about P maps V onto 46

47 47