Assignment Guide: Chapter 9 Geometry

Transcription

1 Assignment Guide: Chapter 9 Geometry (105) 9.1 Translations Page #7-17 odd, 18, 28, 31, 33 (106) 9.2 Reflections Page #7-12, odd, 33, 37 (107) 9.3 Rotations Page #9-15 odd, 17-20, 29, (108) 9.4 Composition of Isometries Page #7-25 odd, (109) Review: Page 577 #1-20 Worksheets (in assignment guide, page 11) Study for quiz ( ) (110) Quiz: Sections Page 585 #29-40 (111) 9.5 Congruence Transformations Page #6-11, (112) 9.6 Dilations Page #7-33 odd, 37, 41 (113) 9.7 Similarity Transformations Page #5-15, (114) Review: Chapter 9 Page 607 #1-26 Worksheets (in assignment guide, pages 20 21) Study for Test (Chapter 9) (115) TEST: Chapter 9 Page 611 #1-20 This is a guide. Homework is subject to change. Check the chalkboard in class for updates and/or changes. 0

2 Geometry Notes, Section 9.1 Translations Translation: A transformation that moves all points in the same direction and same distance. Also called a glide or a slide Can be expressed as coordinates, parallelograms, or a composition of reflections Translations are isometric! Terminology and Notation T : x, y x a, y b OR,, T x y x a y b, where a and b are constants A transformation maps every point of a figure onto its image and may be described with arrow notation ( ). Prime notation ( ) is sometimes used to identify image points. Example: A translation describes how a figure in a coordinate plane is slid from one place to another. In the diagram at right, each point in ABC was moved 4 units down and 1 unit left. A B C is a translation of ABC. Try these: Choose the correct word in each statement. 1. Translations move ( some / all ) points in a figure the same distance, in the same direction. 2. Translations ( change / preserve ) side lengths of the figure. 3. Translations ( change / preserve ) angle measures of the figure. Rigid Motion: A transformation that preserves a figure s shape is a rigid motion. Try this: Does each transformation appear to be a rigid motion?

3 Geometry Practice, Section 9.1 Translations Tell whether the transformation appears to be a rigid motion. Explain Graph the image of each figure under the given translation The dashed-line figure is a translation image of the solid-line figure. Write a rule to describe each translation

4 To solve the following problems, draw a diagram. 9. You are visiting Washington, D.C. From the American History Museum you walk 5 blocks east and 1 block south to the Air and Space Museum. Then you walk 8 blocks west to the Washington Monument. Where is the Washington Monument in relation to the American History Museum? 10. You and some friends go to a book fair where booths are set out in rows. You buy drinks at the refreshment stand and then walk 8 rows north and 2 rows east to the science fiction booth. Then you walk 1 row south and 2 rows west to the children s book booth. Where is the children s book booth in relation to the refreshment stand? 11. Use the graph at the right. Write three different translation rules for which the image of ΔRST has a vertex at the origin. 12. Use the graph at the right. Write three different translation rules for which the image of ΔBCD has a vertex at the origin. 3

5 Geometry Notes, Section 9.2 Reflections Reflection: a transformation that maps every point P to P ' with line m as the line of reflection. If P is not on line m, then line m is the perpendicular bisector of PP '. If P is on the line m, then P P' (they are the same point). Note: a point and its image lie on opposite sides of the line of reflection, unless the point is on the line of reflection. Reflections are isometric! Terminology and Notation Rm is a reflection in line m R P P ' is the reflection that maps P to P ' m Example: Figure ABCD was reflected over the y-axis to form figure LMNO. Figure LMNO is a reflection of figure ABCD. Note that the orientation of the figure reverses. Try these! 1. Point P has coordinates (3, 4). Rx 1 P P'. Find the coordinates of the image P '. 2. Graph ABC, where 3,4 B 0,1, and C 4,2. Graph and label R A B C x axis ' ' '. A, 3. In the diagram, Rt G R H J, and t t G, R D D. Can you use properties of reflections to prove that GHJ is equilateral? Explain how you know. 4

6 Geometry Practice, Section 9.2 Reflections Copy each figure on graph paper. Then, draw the image by reflection in line m Write the coordinates of the image of each point by reflection in (a) the x axis and (b) the y axis. 7. P 8. R 9. Q 10. V Name the image of each figure by reflection in (a) the x axis and (b) the y axis. 11. SR 12. PT 13. QR 14. TV 15. Copy the pentagon at right onto graph paper. Draw its image by reflection in (a) the x axis and (b) the y axis. 16. Create your own design on graph paper. Then draw its image by reflection in (a) the x axis and (b) the y axis. 5

7 Geometry Notes, Section 9.3 Rotations Rotation: a movement in the direction of a given angle, measured in degrees (full rotation is 360 ) counterclockwise (CCW): positive angle clockwise (CW): negative angle half-turn: HO 180 rotations are isometric! Terminology and Notation ( P) (center, angle) ( P ') OR r ( P) ( P ') (center, angle) Rotation in the Coordinate Plane: When a figure is rotated 90, 180, or 270 about the origin O in a coordinate plane, you can use the following rules., 90,, O r x, y 180, x, y O r x, y 270, y, x O r x, y 360, x, y O r x y y x Try these: Graph the following rotations on the coordinate plane given. 1. r 90,O F 2. r GH 90,O 3. r 360,O FGHI 4. r 270,O FGHI 6

8 Geometry Practice, Section 9.3 Rotations Example 1: State two other names for each rotation. (a) ( O,20) (d) X (, 30) (b) ( X, 75) (e) ( A,180) (c) ( O,50) (f) ( O,500) Example 2: O is the center of regular pentagon ABCDE. State the images of A, B, C, D, and E under each rotation. (a) ( O,144) (b) O ( ) (,144) (, 72) O A O A ( ) (,144) ( ) (, 72) O B O B ( ) (,144) O C O C ( ) (,144) O D O D ( ) (,144) O E O E ( ) (, 72) ( ) (, 72) ( ) (, 72) ( ) (, 72) Example 3: X is the center of the regular octagon shown. Complete each of the following. (a) ( X,45) ( S ) (f) ( ) (,45) X P (b) ( ) (, 45) X U (g) X Q ( ) (, 90) (c) ( ) (,90) X R (h) ( ) (,135) X T (d) ( ) (,135) X V (i) H ( ) ( ) X R (e) ( X, 225) ( W ) 7

9 Example 4: State whether the trapezoid is mapped to the other trapezoid by a reflection, translation, rotation, or half-turn. (a) I to II: (b) I to IV: (c) II to III: (d) III to I: (e) II to IV: Example 5: In the diagram, WXYZ is a parallelogram, and T is the midpoint of the diagonals. Can you use properties of rotations to prove that WXYZ is a rhombus? Explain. Example 6: ABCD has vertices A(1, 1), B(1, 3), C(4, 3), and D(4, 1). (a) Graph r (90º, O) (ABCD). (b) Graph r (180º, O) (ABCD). (c) Graph r (270º, O) (ABCD). 8

10 Geometry Notes, Section 9.4 Composition of Isometries Isometry: a transformation that preserves distance, or length Example: Translations, reflections, rotations, and glide reflections are isometries. Note: The composition (combination) of two or more isometries is an isometry. Reflections Across Parallel Lines A composition of reflections across two parallel lines is a translation. You can write this composition as R R ABC A B C m l " " " OR m l " " ". R R ABC A B C The Composition has the following property: AA ", AA ", and AA " are all perpendicular to lines l and m. Try these: Identify each mapping as a translation, reflection, rotation, or glide reflection. Write the rule for each translation, reflection, rotation, or glide reflection. For glide reflections, write the rule as a composition of a translation and a reflection. 1. trapezoid ABCD trapezoid JICD 2. trapezoid ABCD trapezoid NKLM 3. trapezoid CIJD trapezoid LKNM 4. trapezoid CIJD trapezoid TSNU 5. trapezoid KLMN trapezoid STUN 9

11 Geometry Practice, Section 9.4 Composition of Isometries Graph DML and its glide reflection image. 1. (R y-axis T <3, 0> )( DML) 2. (R y=1 T < 1, 0> )( DML) 3. (R x=2 T <0, 1> )( DML) 4. (R y=x T <3, 3> )( DML) Identify each mapping as a translation, reflection, rotation, or glide reflection. Write the rule for each translation, reflection, rotation, or glide reflection. For glide reflections, write the rule as a composition of a translation and a reflection. 5. ABC DEF 6. DEF GHF 7. DEF IJK 8. GHF IJK 10

12 Geometry Review Review: Reflections, Translations, and Glide Reflections Write the coordinates of the image of each point by reflection in the x-axis. 1. H = 2. J = 3. K = 4. L = 5. G = 6. I = Write the coordinates of the image of each point by reflection in the line x = A = 8. B = 9. C = 10. D = 11. E = 12. F = Complete. 13. The translation T : x, y x 2, y 1 The image of glides points 2 units right and. 2,3 is, and the preimage of 2,3 is. 14. If T : 0,0 2,5, then T : 4, If T : 3,4 1,2, then T : 0,0 16. If T : 4, 2 1,1, then T : 2, If T : 3, 4 1, 2, then T : 1,1 In Exercises 17 and 18 a glide reflection is described. Give the coordinates of: (a) A B C, the image of ABC under the glide, and (b) A B C, the image of ABC under the reflection. 17. Glide: All points move up 3 units. Reflection: All points are reflected in the line y = x. A 2,0, B 1,4, C 3, 1 (a) A = B = C = (b) A = B = C = 18. Glide: All points move left 2 units. Reflection: All points are reflected in the y-axis. A 2,5, B 2,4, C 1, 2 (a) A = B = C = (b) A = B = C =.. 11

13 Geometry Notes, Section 9.5 Congruence Transformations Congruent Figures: Two figures are congruent if and only if there is a sequence of one or more rigid motions that maps one figure onto another. Example If a triangle is transformed by the composition of a reflection and a translation, the image is congruent to the given triangle. Try these! 1. The composition (R t T <2,3> ) ( ABC) = XYZ. List all of the equal angle pairs and all of the equal side lengths. 2. Which pairs of figures in the grid are congruent? For each pair, what is a sequence of rigid motions that map one figure to the other? 3. What is a congruence transformation that maps NAV to BCY? 12

14 Geometry Practice, Section 9.5 Congruence Transformations For each coordinate grid, identify a pair of congruent figures. Then determine a congruence transformation that maps the preimage to the congruent image Find a congruence transformation that maps ABC to DEF Error Analysis 6. For the figure at the right, a student says that ΔABC ΔDEF because R y-axis (ΔABC) = ΔDEF. What is the student s error? 13

15 Geometry Notes, Section 9.6 Dilations Dilation: a transformation that maps any segment to a parallel segment k times as long. k is called the scale factor an expansion (or enlargement) is when the image is larger than the preimage, k 1 a contraction (or reduction) is when the preimage is larger than the image, k 1 dilations are not isometric, they are similarity mappings same shape but not necessarily the same size Terminology and Notation D ( ) (center, ) P P ' k If k 0, then P ' lies on OP and OP ' k OP. If k 0, then P ' lies on the ray opposite OP and OP ' k OP. A dilation with center of dilation C and scale factor n, n > 0, can be written as D (n, C). A dilation is a transformation with the following properties: The image of C is itself (that is, C = C). For any other point R, R is on CR and CR = n CR, or n = CR /CR. Dilations preserve angle measure. Try these! 1. What are the coordinates of the vertices of D ½ ( PZG)? 2. The height of a document on your computer screen is 20.4 cm. When you change the zoon setting on your screen from 100% to 25%, the new image of your document is a dilation of the previous image with scale factor What is the height of the new image? 14

16 Geometry Practice, Section 9.6 Dilations Find the image of ABC under the dilation. Is the dilation an expansion or contraction? 1. O,3 D 2. O, 1 3 A 1,0, B 2,2, C 3, 2 A 3,0, B 6,9, C 3, 6 D Find the coordinates of the images of P 4,0, Q 4,8, and R 2, 4 Tell if the dilation is an expansion or a contraction. 3. O,4 under the given dilation. D 4. DO, 4 5. D O, O, 1 2 D 7. DO, 1 8. D O,2 A dilation with the origin O maps the given point to the image point named. Find the scale factor of the dilation. Is the dilation an expansion or contraction? 9. 3,0 15, ,3 8,2 A dilation with the origin O maps the given point to the image point named. Find the scale factor of the dilation. Tell if the dilation is an expansion or a contraction ,0 10, ,8 2, ,5 16, ,4, , 0, 16. 3,8 9,

17 Geometry Practice, Sections 9.3 & 9.6 Rotations and Dilations 16

18 Geometry Notes, Section 9.7 Similarity Transformations Similar Figures: Two figures are similar if and only if there is a similarity transformation that maps one figure onto another. Example When a figure is transformed by a similarity transformation (a composition of rigid motion and dilation), a similar figure is formed. Try these! LMN has vertices L(-4, 2), M(-3, -3), and N(-1, 1). Suppose the triangle is translated 4 units right and 2 units up and then dilated by a scale factor of 0.5 with center of dilation at the origin. Sketch the resulting image of the composition of transformations. 2. What is a composition of rigid motions and a dilation that maps trapezoid ABCD to trapezoid MNHP? 4. For TUV shown at right, give the vertices of a similar triangle after a similarity transformation that uses at least one rigid motion. 5. Write the coordinates of the reflection of TUV across the y-axis. 6. Draw the dilation of T U V using a scale factor of 0.5 and the origin as the center of the dilation. 17

19 Geometry Practice, Section 9.7 Similarity Transformations 7. A transformation maps each point (x, y) of ΔABC to the point ( 4x + 3, 4y + 2). Is ΔABC similar to the image of ΔABC? Explain. 18

20 Geometry Review Review Write a congruence or similarity statement for the two figures in each coordinate grid. Then write a congruence transformation or similarity transformation that maps one figure to the other Use the diagram at right for # 5 6. The solid-line figure is a dilation of the dashed-line figure with center of dilation M. 5. Is the dilation an enlargement or a reduction? 6. What is the scale factor of the dilation? 7. Given P(3, 11), what are the coordinates of D 3 (P)? 8. Given P(2, 4), what are the coordinates of D 4 (P)? 19

21 Geometry Review Chapter 9 Review R S T is a translation image of RST. 1. What is a rule for the translation? 2. Is a glide reflection a rigid motion? Explain. Find the coordinates of the vertices of each image. 3. R y-axis (QRST) 4. r (270, O) (QRST) 5. D 5 (QRST) 6. T <2, 5> (QRST) 7. (R y = 2 T < 4, 0> )(QRST) Write a single transformation rule that has the same effect as each composition of transformations. 8. T <6, 7> T < 10, 7> 9. R x = 6 Rx = R y = 2 T <0, 1> Identify the rigid motion that maps the solid-line figure onto the dotted-line figure Find the image of each point for the given dilation. 13. L (6, 4); D 1. 2 (L) 14. S( 2, 6); D (S) 15. W( 3, 2); D 5 (W) 20

22 Write a congruence or similarity statement for the two figures in each coordinate grid. Then write a congruence transformation or similarity transformation that maps one figure to the other Lines a and b are parallel. The letter A is reflected first across line a so that the image is between lines a and b. The figure is then reflected across line b. Describe the image of A. 21. What is the image of ( 6, 6) after a reflection across the y-axis? 22. ABC has vertices at A(0, 5), B(4, 4), and C( 1, 0). Graph the image T < 1, 2> ( ABC). 21