To draw and identify rotation images of figures

Transcription

1 9-3 otations ommon ore State Standards G-.. evelop definitions of rotations... in terms of angles, circles, perpendicular lines, parallel lines, and line segments. lso G-.., G-..6 M 1, M 3, M bjective o draw and identify rotation images of figures Notice the position of the point, in relation to the - and y-ais, as it rotates around the origin. In the diagram, the point (3, ) is rotated counterclockwise about the origin. he point ( 1, y 1 ) is the result of a 90 rotation. he point (, y ) is the result of a 180 rotation, and the point ( 3, y 3 ) is the result of a 70 rotation. What are the coordinates of ( 1, y 1 ), (, y ), and ( 3, y 3 )? What do you notice about how the coordinates of the points relate to the coordinates (3, ) after each rotation? ( 1, y 1 ) (, y ) y (3, ) ( 3, y 3 ) MHMI IS In the Solve It, you thought about how the coordinates of a point change as it turns, or rotates, about the origin on a coordinate grid. In this lesson, you will learn how to recognize and construct rotations of geometric figures. ssential Understanding otations preserve distance, angle measures, and orientation of figures. esson VocabularyV rotation center of rotation angle of rotation ey oncept otation bout a oint rotation of about a point, called the center of rotation, is a transformation with these two properties: he image of is itself (that is, = ). For any other point V, V = V and m VV =. he number of degrees a figure rotates is the angle of rotation. rotation about a point is a rigid motion. You write the rotation of UVW about point as r (, ) ( UVW) = U V W. V V W W U U he preimage V and its image V are equidistant from the center of rotation. Unless stated otherwise, rotations in this book are counterclockwise. esson 9-3 otations 561

2 roblem 1 rawing a otation Image What is the image of r (100, ) ( )? How do you use the definition of rotation about a point to help you get started? You know that and 9 must be equidistant from and that m must be 100. Step 1 raw. Use a protractor to draw a 100 angle with verte and side. 100 Step Use a compass to construct. Step 3 ocate 9 and 9 in a similar manner. Step raw. Got It? 1. opy from roblem 1. What is the image of for a 50 rotation about? When a figure is rotated 90, 180, or 70 about the origin in a coordinate plane, you can use the following rules. ey oncept otation in the oordinate lane r (90, ) (, y) = (-y, ) G ( 3, ) 6 G (, 3) 6 r (180, ) (, y) = (-, -y) G (, 3) G (, 3) r (70, ) (, y) = (y, -) r (360, ) (, y) = (, y) G (, 3) G (, 3) G (3, ) 56 hapter 9 ransformations

3 How do you know where to draw the vertices on the coordinate plane? Use the rules for rotating a point and apply them to each verte of the figure. hen graph the points and connect them to draw the image. roblem rawing otations in a oordinate lane S has vertices (1, 1), (3, 3), (, 1), and S(3, 0). What is the graph of r (90, ) (S). First, graph the images of each verte. = r (90, ) (1, 1) = (-1, 1) = r (90, ) (3, 3) = (-3, 3) = r (90, ) (, 1) = (-1, ) S = r (90, ) (3, 0) = (0, 3) Net, connect the vertices to graph S. ( 1, ) ( 3, 3) S y (0, 3) ( 1, 1) 6 S 6 Got It?. Graph r (70, ) (FGHI). F ( 3, ) 6 G ( 3, 1) I (0, 1) H ( 1, 1) 6 You can use the properties of rotations to solve problems. roblem 3 Using roperties of otations What do you know about rotations that can help you show that opposite sides of the parallelogram are equal? You know that rotations are rigid motions, so if you show that the opposite sides can be mapped to each other, then the side lengths must be equal. In the diagram, WXYZ is a parallelogram, and is the midpoint of the diagonals. How can you use the properties of rotations to show that the lengths of the opposite sides of the parallelogram are equal? ecause is the midpoint of the diagonals, X = Z and W = Y. Since W and Y are equidistant from, and the measure of WY = 180, you know that r (180, ) (W) = Y. Similarly, r (180, ) (X) = Z. You can rotate every point on WX in this same way, so r (180, ) (WX) = YZ. ikewise, you can map WZ to YX with r (180, ) (WZ) = YX. ecause rotations are rigid motions and preserve distance, WX = YZ and WZ = YX. Got It? 3. an you use the properties of rotations to prove that WXYZ is a rhombus? plain. X W Y Z esson 9-3 otations 563

4 esson heck o you know HW? 1. opy the figure and point. raw r (70, ) ( ). In the figure below, point is the center of square S.. What is r (90, ) ()? 3. What is the image of for a 180 rotation about?. Use the properties of rotations to describe how you know that the lengths of the diagonals of the square are equal. S o you UNSN? 5. Vocabulary is a rotation image of about point. escribe how to find the angle of rotation. 6. rror nalysis classmate drew a 115 rotation of about point, as shown at the right. plain and correct your classmate s error. MHMI IS 7. ompare and ontrast ompare rotating a figure about a point to reflecting the figure across a line. How are the transformations alike? How are they different? 8. easoning oint (, y) is rotated about the origin by 135 and then by 5. What are the coordinates of the image of point? plain 115 ractice and roblem-solving ercises MHMI IS ractice opy each figure and point. raw the image of each figure for the given rotation about. Use prime notation to label the vertices of the image. See roblem opy each figure and point. hen draw the image of J for a 180 rotation about. Use prime notation to label the vertices of the image J 16. J J J 56 hapter 9 ransformations

5 For ercises 17 19, use the graph at the right. 17. Graph r (90, ) (FGHJ). 18. Graph r (180, ) (FGHJ). 19. Graph r (70, ) (FGHJ). 0. he coordinates of S are (-3, ), (, 5), and S(0, 0). What are the coordinates of the vertices of r (70, ) ( S)? F (0, 3) G (, 1) 1. V W X Y has vertices V (-3, ), W (5, 1), X (0, ), and Y (-, 0). If r (90, ) (VWXY) = V W X Y, what are the coordinates of VWXY?. Ferris Wheel Ferris wheel is drawn on a coordinate plane so that the first car is located at the point (30, 0). What are the coordinates of the first car after a rotation of 70 about the origin? For ercises 3 5, use the diagram at the right. NV is a rectangle. M is the midpoint of the diagonals. 6 J (3, ) H (1, ) 6 See roblem. See roblem Use the properties of rotations to show that the measures of both pairs of opposite sides are equal in length.. easoning an you use the properties of rotations to show that the measures of the lengths of the diagonals are equal? V M N 5. easoning an you use properties of rotations to conclude that the diagonals of NV bisect the angles of NV? plain. pply 6. In the diagram at the right, M N is the rotation image of MN about point. Name all pairs of angles and all pairs of segments that have equal measures in the diagram. 7. anguage rts Symbols are used in dictionaries to help users pronounce words correctly. he symbol is called a schwa. It is used in dictionaries to represent neutral vowel sounds such as a in ago, i in sanity, and u in focus. What transformation maps a to a lowercase e? M N M N Find the angle of rotation about that maps the black figure to the blue figure esson 9-3 otations 565

6 31. hink bout a lan he Millenium Wheel, also known as the ondon ye, contains 3 observation cars. etermine the angle of rotation that will bring ar 3 to the position of ar 18. How do you find the angle of rotation that a car travels when it moves one position counterclockwise? How many positions does ar 3 move? ar 3 3. easoning For center of rotation, does an rotation followed by a y rotation give the same image as a y rotation followed by an rotation? plain. 33. Writing escribe how a series of rotations can have the same effect as a 360 rotation about a point X. 3. oordinate Geometry Graph (5, ). Graph, the image of for a 90 rotation about the origin. Graph, the image of for a 180 rotation about. Graph, the image of for a 70 rotation about. What type of quadrilateral is? plain. ar 18 oint is the center of the regular nonagon shown at the right. 35. Find the angle of rotation that maps F to H. I 36. pen-nded escribe a rotation that maps H to. 37. rror nalysis Your friend says that is the image of for a 10 rotation about. What is wrong with your friend s statement? H G F In the figure at the right, the large triangle, the quadrilateral, and the heagon are regular. Find the image of each point or segment for the given rotation or composition of rotations. (Hint: djacent green segments form 30 angles.) I J 38. r (10, ) () 39. r (70, ) () 0. r (300, ) (I) 1. r (60, ) () H M. r (180, ) (J) 3. r (0, ) (G) hallenge. r (10, H) (F) 5. r (70, ) (M) G 6. r (180, ) (I) 7. r (70, ) (M) F 8. oordinate Geometry raw MN with vertices (, -1), M(6, -), and N(, ). Find the coordinates of the vertices after a 90 rotation about the origin and about each of the points, M, and N. 9. easoning If you are given a figure and a rotation image of the figure, how can you find the center and angle of rotation? 566 hapter 9 ransformations

7 Standardized est rep S/ 50. What is the image of (1, -6) for a 90 counterclockwise rotation about the origin? (6, 1) (-1, 6) (-6, -1) (-1, -6) 51. he costume crew for your school musical makes aprons like the one shown. If blue ribbon costs $1.50 per foot, what is the cost of ribbon for si aprons? $15.75 $.00 $31.50 $ in. in. 5. In, m + m = 8. Which statement must be true? Short esponse 53. Use the following statement: If two lines are parallel, then the lines do not intersect. a. What are the converse, inverse, and contrapositive of the statement? b. What is the truth value of each statement you wrote in part (a)? If a statement is false, give a countereample. FMN S pply What You ve earned ook back at the information about the video game on page 53. he graph of the puzzle piece and target area is shown again below. MHMI IS M 5 y F In the pply What You ve earned in esson 9-, you looked at how some translations and reflections move the puzzle piece. Now you will look at how rotations move the puzzle piece. a. How does the orientation of the puzzle piece compare to the orientation of the target area? b. an you move the puzzle piece to the target area using only rotations? plain. For parts (c) (e), copy the graph and then graph the image of for the given rotation. c. r (90, ) (, y) d. r (180, ) (, y) e. r (70, ) (, y) esson 9-3 otations 567