4 Transformations 4.1 Translations 4.2 Reflections 4.3 Rotations 4.4 Congruence and Transformations 4.5 Dilations 4.6 Similarity and Transformations

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1 Transformations.1 Translations. Reflections.3 Rotations. ongruence and Transformations.5 ilations.6 Similarit and Transformations hapter Learning Target: Understand transformations. hapter Success riteria: I can identif transformations. I can perform translations, reflections, rotations, and dilations. I can describe congruence and similarit transformations. I can solve problems involving transformations. Magnification (p. 13) hoto Stickers (p. 11) Kaleidoscope (p. 196) S the ig Idea hess (p. 179) Revolving oor (p. 195)

2 Maintaining Mathematical roficienc Identifing Transformations ample 1 Tell whether the red figure is a translation, reflection, rotation, or dilation of the blue figure. a. The blue figure b. The red figure is a turns to form mirror image of the the red figure, blue figure, so it is so it is a rotation. a reflection. Tell whether the red figure is a translation, reflection, rotation, or dilation of the blue figure Identifing Similar igures ample Which rectangle is similar to Rectangle? Rectangle 8 Rectangle ach figure is a rectangle, so corresponding angles are congruent. heck to see whether corresponding side lengths are proportional. 1 Rectangle 6 3 Rectangle and Rectangle Rectangle and Rectangle Length of Length of = 8 = Width of Width of = 1 = Length of Length of = 8 6 = 3 not proportional So, Rectangle is similar to Rectangle. proportional Width of Width of = 3 Tell whether the two figures are similar. plain our reasoning STRT RSONING an ou draw two squares that are not similar? plain our reasoning. namic Solutions available at igideasmath.com 171

3 Mathematical ractices Using namic Geometr Software ore oncept Using namic Geometr Software Mathematicall profi cient students use dnamic geometr software strategicall. namic geometr software allows ou to create geometric drawings, including: drawing a point drawing a line drawing a line segment drawing an angle measuring an angle measuring a line segment drawing a circle drawing an ellipse drawing a perpendicular line drawing a polgon coping and sliding an object reflecting an object in a line Monitoring rogress inding Side Lengths and ngle Measures Use dnamic geometr software to draw a triangle with vertices at (, 1), (, 1), and (, ). ind the side lengths and angle measures of the triangle. Using dnamic geometr software, ou can create, as shown Sample oints (, 1) (, 1) (, ) Segments = = 3 = 5 ngles m = m = 90 m = rom the displa, the side lengths are = units, = 3 units, and = 5 units. The angle measures, rounded to two decimal places, are m 36.87, m = 90, and m Use dnamic geometr software to draw the polgon with the given vertices. Use the software to find the side lengths and angle measures of the polgon. Round our answers to the nearest hundredth. 1. (0, ), (3, 1), (, 3). (, 1), (, 1), (3, ) 3. (1, 1), ( 3, 1), ( 3, ), (1, ). (1, 1), ( 3, 1), (, ), (, ) 5. ( 3, 0), (0, 3), (3, 0), (0, 3) 6. (0, 0), (, 0), (1, 1), (0, 3) 17 hapter Transformations

4 .1 Translations ssential Question How can ou translate a figure in a coordinate plane? Translating a Triangle in a oordinate lane USING TOOLS STRTGILLY To be proficient in math, ou need to use appropriate tools strategicall, including dnamic geometr software. Work with a partner. a. Use dnamic geometr software to draw an triangle and label it. b. op the triangle and translate (or slide) it to form a new figure, called an image, (read as triangle prime, prime, prime ). c. What is the relationship between the coordinates of the vertices of and those of? d. What do ou observe about the side lengths and angle measures of the two triangles? Sample oints ( 1, ) (3, ) (, 1) Segments = = 3.16 =. ngles m = 5 m = m = 63.3 Translating a Triangle in a oordinate lane Work with a partner. a. The point (, ) is translated a units horizontall and b units verticall. Write a rule to determine the coordinates of the image of (, ). (, ) (, ) b. Use the rule ou wrote in part (a) to translate units left and 3 units down. What are the coordinates of the vertices of the image,? c. raw. re its side lengths the same as those of? Justif our answer. omparing ngles of Translations Work with a partner. a. In ploration, is a right triangle? Justif our answer. b. In ploration, is a right triangle? Justif our answer. c. o ou think translations alwas preserve angle measures? plain our reasoning. ommunicate Your nswer. How can ou translate a figure in a coordinate plane? 5. In ploration, translate 3 units right and units up. What are the coordinates of the vertices of the image,? How are these coordinates related to the coordinates of the vertices of the original triangle,? Section.1 Translations 173

5 .1 Lesson ore Vocabular vector, p. 17 initial point, p. 17 terminal point, p. 17 horizontal component, p. 17 vertical component, p. 17 component form, p. 17 transformation, p. 17 image, p. 17 preimage, p. 17 translation, p. 17 rigid motion, p. 176 composition of transformations, p. 176 What You Will Learn erform translations. erform compositions. Solve real-life problems involving compositions. erforming Translations vector is a quantit that has both direction and magnitude, or size, and is represented in the coordinate plane b an arrow drawn from one point to another. ore oncept Vectors The diagram shows a vector. The initial point, or starting point, of the vector is, and the terminal point, or ending point, is Q. The vector is named Q, which is read as vector Q. The horizontal component of Q is 5, and the vertical component is 3. The component form of a vector combines the horizontal and vertical components. So, the component form of Q is 5, 3. 5 units right Q 3 units up J K Identifing Vector omponents In the diagram, name the vector and write its component form. The vector is JK. To move from the initial point J to the terminal point K, ou move 3 units right and units up. So, the component form is 3,. transformation is a function that moves or changes a figure in some wa to produce a new figure called an image. nother name for the original figure is the preimage. The points on the preimage are the inputs for the transformation, and the points on the image are the outputs. STUY TI You can use prime notation to name an image. or eample, if the preimage is point, then its image is point, read as point prime. ore oncept Translations translation moves ever point of a figure the same distance in the same direction. More specificall, ( 1, 1 ) a translation maps, or moves, the points and Q of a plane figure along a vector a, b to the points and Q, so that one of the following Q(, ) statements is true. = QQ and QQ, or = QQ and and QQ are collinear. ( 1 + a, 1 + b) Q ( + a, + b) Translations map lines to parallel lines and segments to parallel segments. or instance, in the figure above, Q Q. 17 hapter Transformations

6 Translating a igure Using a Vector The vertices of are (0, 3), (, ), and (1, 0). Translate using the vector 5, 1. irst, graph. Use 5, 1 to move each verte 5 units right and 1 unit down. Label the image vertices. raw. Notice that the vectors drawn from preimage vertices to image vertices are parallel. (5, ) (7, 3) 8 (6, 1) You can also epress a translation along the vector a, b using a rule, which has the notation (, ) ( + a, + b). Writing a Translation Rule Write a rule for the translation of to To go from to, ou move units left and 1 unit up, so ou move along the vector, 1. So, a rule for the translation is (, ) (, + 1). Translating a igure in the oordinate lane Graph quadrilateral with vertices ( 1, ), ( 1, 5), (, 6), and (, ) and its image after the translation (, ) ( + 3, 1). 6 Graph quadrilateral. To find the coordinates of the vertices of the image, add 3 to the -coordinates and subtract 1 from the -coordinates of the vertices of the preimage. Then graph the image, as shown at the left. (, ) ( + 3, 1) 6 ( 1, ) (, 1) ( 1, 5) (, ) (, 6) (7, 5) (, ) (7, 1) Monitoring rogress Help in nglish and Spanish at igideasmath.com 1. Name the vector and write its component form. K. The vertices of LMN are L(, ), M(5, 3), and N(9, 1). Translate LMN using the vector, In ample 3, write a rule to translate back to.. Graph RST with vertices R(, ), S(5, ), and T(3, 5) and its image after the translation (, ) ( + 1, + ). Section.1 Translations 175

7 erforming ompositions rigid motion is a transformation that preserves length and angle measure. nother name for a rigid motion is an isometr. rigid motion maps lines to lines, ras to ras, and segments to segments. ostulate ostulate.1 Translation ostulate translation is a rigid motion. ecause a translation is a rigid motion, and a rigid motion preserves length and angle measure, the following statements are true for the translation shown. =, =, = m = m, m = m, m = m When two or more transformations are combined to form a single transformation, the result is a composition of transformations. Theorem Theorem.1 omposition Theorem The composition of two (or more) rigid motions is a rigid motion. roof. 35, p. 180 Q The theorem above is important because it states that no matter how man rigid motions ou perform, lengths and angle measures will be preserved in the final image. or instance, the composition of two or more translations is a translation, as shown. composition Q translation 1 translation Q erforming a omposition Graph RS with endpoints R( 8, 5) and S( 6, 8) and its image after the composition. Translation: (, ) ( + 5, ) Translation: (, ) (, ) Step 1 Graph RS. Step Translate RS 5 units right and units down. R S has endpoints R ( 3, 3) and S ( 1, 6). Step 3 Translate R S units left and units down. R S has endpoints R ( 7, 1) and S ( 5, ). S( 6, 8) S ( 1, 6) 8 6 R( 8, 5) S ( 5, ) R ( 3, 3) R ( 7, 1) hapter Transformations

8 Solving Real-Life roblems Modeling with Mathematics You are designing a favicon for a golf website. In an image-editing program, ou move the red rectangle units left and 3 units down. Then ou move the red rectangle 1 unit right and 1 unit up. Rewrite the composition as a single translation Understand the roblem You are given two translations. You need to rewrite the result of the composition of the two translations as a single translation.. Make a lan You can choose an arbitrar point (, ) in the red rectangle and determine the horizontal and vertical shift in the coordinates of the point after both translations. This tells ou how much ou need to shift each coordinate to map the original figure to the final image. 3. Solve the roblem Let (, ) be an arbitrar point in the red rectangle. fter the first translation, the coordinates of its image are (, 3). The second translation maps (, 3) to ( + 1, 3 + 1) = ( 1, ). The composition of translations uses the original point (, ) as the input and returns the point ( 1, ) as the output. So, the single translation rule for the composition is (, ) ( 1, ).. Look ack heck that the rule is correct b testing a point. or instance, (10, 1) is a point in the red rectangle. ppl the two translations to (10, 1). (10, 1) (8, 9) (9, 10) oes the final result match the rule ou found in Step 3? (10, 1) (10 1, 1 ) = (9, 10) Monitoring rogress Help in nglish and Spanish at igideasmath.com 5. Graph TU with endpoints T(1, ) and U(, 6) and its image after the composition. Translation: (, ) (, 3) Translation: (, ) (, + 5) 6. Graph VW with endpoints V( 6, ) and W( 3, 1) and its image after the composition. Translation: (, ) ( + 3, + 1) Translation: (, ) ( 6, ) 7. In ample 6, ou move the gra square units right and 3 units up. Then ou move the gra square 1 unit left and 1 unit down. Rewrite the composition as a single transformation. Section.1 Translations 177

9 .1 ercises namic Solutions available at igideasmath.com Vocabular and ore oncept heck 1. VOULRY Name the preimage and image of the transformation.. OMLT TH SNTN moves ever point of a figure the same distance in the same direction. Monitoring rogress and Modeling with Mathematics In ercises 3 and, name the vector and write its component form. (See ample 1.) M M L L N 5 N. S In ercises 13 16, use the translation. (, ) ( 8, + ) 13. What is the image of (, 6)? T In ercises 5 8, the vertices of are (, 5), (6, 3), and (, 0). Translate using the given vector. Graph and its image. (See ample.) 5. 6, , , 7 8., In ercises 9 and 10, find the component form of the vector that translates ( 3, 6) to. 9. (0, 1) 10. (, 8) In ercises 11 and 1, write a rule for the translation of LMN to L M N. (See ample 3.) 11. L M N L M 6 N 1. What is the image of ( 1, 5)? 15. What is the preimage of ( 3, 10)? 16. What is the preimage of (, 3)? In ercises 17 0, graph QR with vertices (, 3), Q(1, ), and R(3, 1) and its image after the translation. (See ample.) 17. (, ) ( +, + 6) 18. (, ) ( + 9, ) 19. (, ) (, 5) 0. (, ) ( 1, + 3) In ercises 1 and, graph XYZ with vertices X(, ), Y(6, 0), and Z(7, ) and its image after the composition. (See ample 5.) 1. Translation: (, ) ( + 1, + ) Translation: (, ) ( 5, 9). Translation: (, ) ( 6, ) Translation: (, ) ( +, + 7) 178 hapter Transformations

10 In ercises 3 and, describe the composition of translations. 3.. G G 1 3 G 5 5. RROR NLYSIS escribe and correct the error in graphing the image of quadrilateral GH after the translation (, ) ( 1, ) H G H G 9 6. MOLING WITH MTHMTIS In chess, the knight (the piece shaped like a horse) moves in an L pattern. The board shows two consecutive moves of a black knight during a game. Write a composition of translations for the moves. Then rewrite the composition as a single translation that moves the knight from its original position to its ending position. (See ample 6.) 7. ROLM SOLVING You are studing an amoeba through a microscope. Suppose the amoeba moves on a grid-indeed microscope slide in a straight line from square 3 to square G7. GH X 8 a. escribe the translation. b. The side length of each grid square is millimeters. How far does the amoeba travel? c. The amoeba moves from square 3 to square G7 in.5 seconds. What is its speed in millimeters per second? 8. MTHMTIL ONNTIONS Translation maps (, ) to ( + n, + t). Translation maps (, ) to ( + s, + m). a. Translate a point using Translation, followed b Translation. Write an algebraic rule for the final image of the point after this composition. b. Translate a point using Translation, followed b Translation. Write an algebraic rule for the final image of the point after this composition. c. ompare the rules ou wrote for parts (a) and (b). oes it matter which translation ou do first? plain our reasoning. MTHMTIL ONNTIONS In ercises 9 and 30, a translation maps the blue figure to the red figure. ind the value of each variable w r s t 10 a b c 6 8 Section.1 Translations 179

11 31. USING STRUTUR Quadrilateral G has vertices ( 1, ), (, 0), ( 1, 1), and G(1, 3). translation maps quadrilateral G to quadrilateral G. The image of is (, ). What are the coordinates of,, and G? 3. HOW O YOU S IT? Which two figures represent a translation? escribe the translation. 35. ROVING THORM rove the omposition Theorem (Theorem.1). 36. ROVING THORM Use properties of translations to prove each theorem. a. orresponding ngles Theorem (Theorem 3.1) b. orresponding ngles onverse (Theorem 3.5) 37. WRITING plain how to use translations to draw a rectangular prism MTHMTIL ONNTIONS The vector Q =, 1 describes the translation of ( 1, w) onto ( + 1, ) and (8 1, 1) onto (3, 3z). ind the values of w,,, and z. 39. MKING N RGUMNT translation maps GH to G H. Your friend claims that if ou draw segments connecting G to G and H to H, then the resulting quadrilateral is a parallelogram. Is our friend correct? plain our reasoning. 33. RSONING The translation (, ) ( + m, + n) maps Q to Q. Write a rule for the translation of Q to Q. plain our reasoning. 3. RWING ONLUSIONS The vertices of a rectangle are Q(, 3), R(, ), S(5, ), and T(5, 3). a. Translate rectangle QRST 3 units left and 3 units down to produce rectangle Q R S T. ind the area of rectangle QRST and the area of rectangle Q R S T. b. ompare the areas. Make a conjecture about the areas of a preimage and its image after a translation. 0. THOUGHT ROVOKING You are a graphic designer for a compan that manufactures floor tiles. esign a floor tile in a coordinate plane. Then use translations to show how the tiles cover an entire floor. escribe the translations that map the original tile to four other tiles. 1. RSONING The vertices of are (, ), (, ), and (3, ). Graph the image of after the transformation (, ) ( +, ). Is this transformation a translation? plain our reasoning.. ROO MN is perpendicular to line. M N is the translation of MN units to the left. rove that M N is perpendicular to. Maintaining Mathematical roficienc Tell whether the figure can be folded in half so that one side matches the other. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons Simplif the epression. (Skills Review Handbook) 7. ( ) 8. ( + 3) 9. (1 5) 50. ( + ) 180 hapter Transformations

12 . Reflections ssential Question How can ou reflect a figure in a coordinate plane? Reflecting a Triangle Using a Reflective evice Work with a partner. Use a straightedge to draw an triangle on paper. Label it. a. Use the straightedge to draw a line that does not pass through the triangle. Label it m. b. lace a reflective device on line m. c. Use the reflective device to plot the images of the vertices of. Label the images of vertices,, and as,, and, respectivel. d. Use a straightedge to draw b connecting the vertices. LOOKING OR STRUTUR To be proficient in math, ou need to look closel to discern a pattern or structure. Reflecting a Triangle in a oordinate lane Work with a partner. Use dnamic geometr software to draw an triangle and label it. a. Refl ect in the -ais to form. b. What is the relationship between the coordinates of the vertices of and those of? c. What do ou observe about the side lengths and angle measures of the two triangles? d. Refl ect in the -ais to form. Then repeat parts (b) and (c). Sample oints ( 3, 3) 3 (, 1) ( 1, ) Segments =.1 1 = = ngles 1 m = m = 5.35 m = 5.13 ommunicate Your nswer 3. How can ou reflect a figure in a coordinate plane? Section. Reflections 181

13 . Lesson What You Will Learn ore Vocabular reflection, p. 18 line of reflection, p. 18 glide reflection, p. 18 line smmetr, p. 185 line of smmetr, p. 185 erform reflections. erform glide reflections. Identif lines of smmetr. Solve real-life problems involving reflections. erforming Reflections ore oncept Reflections reflection is a transformation that uses a line like a mirror to reflect a figure. The mirror line is called the line of reflection. reflection in a line m maps ever point in the plane to a point, so that for each point one of the following properties is true. If is not on m, then m is the perpendicular bisector of, or m m If is on m, then =. point not on m point on m Reflecting in Horizontal and Vertical Lines Graph with vertices (1, 3), (5, ), and (, 1) and its image after the reflection described. a. In the line n: = 3 b. In the line m: = 1 a. oint is units left of line n, so its reflection is units right of line n at (5, 3). lso, is units left of line n at (1, ), and is 1 unit right of line n at (, 1). b. oint is units above line m, so is units below line m at (1, 1). lso, is 1 unit below line m at (5, 0). ecause point is on line m, ou know that =. n 6 6 m Monitoring rogress Help in nglish and Spanish at igideasmath.com Graph from ample 1 and its image after a reflection in the given line. 1. =. = 3 3. =. = 1 18 hapter Transformations

14 Reflecting in the Line = Graph G with endpoints ( 1, ) and G(1, ) and its image after a reflection in the line =. RMMR The product of the slopes of perpendicular lines is 1. The slope of = is 1. The segment from to its image,, is perpendicular to the line of reflection =, so the slope of will be 1 (because 1( 1) = 1). rom, move 1.5 units right and 1.5 units down to =. rom that point, move 1.5 units right and 1.5 units down to locate (, 1). The slope of GG will also be 1. rom G, move 0.5 unit right and 0.5 unit down to =. Then move 0.5 unit right and 0.5 unit down to locate G (, 1). = G G You can use coordinate rules to find the images of points reflected in four special lines. ore oncept oordinate Rules for Reflections If (a, b) is reflected in the -ais, then its image is the point (a, b). If (a, b) is reflected in the -ais, then its image is the point ( a, b). If (a, b) is reflected in the line =, then its image is the point (b, a). If (a, b) is reflected in the line =, then its image is the point ( b, a). Reflecting in the Line = Graph G from ample and its image after a reflection in the line =. Use the coordinate rule for reflecting in the line = to find the coordinates of the endpoints of the image. Then graph G and its image. (a, b) ( b, a) ( 1, ) (, 1) G(1, ) G (, 1) G G = Monitoring rogress The vertices of JKL are J(1, 3), K(, ), and L(3, 1). 5. Graph JKL and its image after a reflection in the -ais. 6. Graph JKL and its image after a reflection in the -ais. Help in nglish and Spanish at igideasmath.com 7. Graph JKL and its image after a reflection in the line =. 8. Graph JKL and its image after a reflection in the line =. 9. In ample 3, verif that is perpendicular to =. Section. Reflections 183

15 erforming Glide Reflections ostulate ostulate. Reflection ostulate reflection is a rigid motion. m ecause a reflection is a rigid motion, and a rigid motion preserves length and angle measure, the following statements are true for the reflection shown. =, =, = m = m, m = m, m = m ecause a reflection is a rigid motion, the omposition Theorem (Theorem.1) guarantees that an composition of reflections and translations is a rigid motion. STUY TI The line of reflection must be parallel to the direction of the translation to be a glide reflection. glide reflection is a transformation involving a translation followed b a reflection in which ever point is mapped to a point b the following steps. Step 1 irst, a translation maps to. Step Then, a reflection in a line k parallel to the direction of the translation maps to. Q Q Q k erforming a Glide Reflection Graph with vertices (3, ), (6, 3), and (7, 1) and its image after the glide reflection. Translation: (, ) ( 1, ) Reflection: in the -ais egin b graphing. Then graph after a translation 1 units left. inall, graph after a reflection in the -ais. ( 6, 3) (6, 3) ( 9, ) ( 9, ) ( 5, 1) 6 ( 5, 1) ( 6, 3) (3, ) (7, 1) 6 8 Monitoring rogress Help in nglish and Spanish at igideasmath.com 10. WHT I? In ample, is translated units down and then reflected in the -ais. Graph and its image after the glide reflection. 11. In ample, describe a glide reflection from to. 18 hapter Transformations

16 Identifing Lines of Smmetr figure in the plane has line smmetr when the figure can be mapped onto itself b a reflection in a line. This line of reflection is a line of smmetr, such as line m at the left. figure can have more than one line of smmetr. m Identifing Lines of Smmetr How man lines of smmetr does each heagon have? a. b. c. a. b. c. Monitoring rogress etermine the number of lines of smmetr for the figure Help in nglish and Spanish at igideasmath.com 15. raw a heagon with no lines of smmetr. Solving Real-Life roblems inding a Minimum istance You are going to bu books. Your friend is going to bu s. Where should ou park to minimize the distance ou both will walk? Reflect in line m to obtain. Then draw. Label the intersection of and m as. ecause is the shortest distance between and and =, park at point to minimize the combined distance, +, ou both have to walk. m m Monitoring rogress Help in nglish and Spanish at igideasmath.com 16. Look back at ample 6. nswer the question b using a reflection of point instead of point. Section. Reflections 185

17 . ercises namic Solutions available at igideasmath.com Vocabular and ore oncept heck 1. VOULRY glide reflection is a combination of which two transformations?. WHIH ON OSN T LONG? Which transformation does not belong with the other three? plain our reasoning. 6 Monitoring rogress and Modeling with Mathematics In ercises 3 6, determine whether the coordinate plane shows a reflection in the -ais, -ais, or neither In ercises 7 1, graph JKL and its image after a reflection in the given line. (See ample 1.) 7. J(, ), K(3, 7), L(6, 1); -ais 8. J(5, 3), K(1, ), L( 3, ); -ais 9. J(, 1), K(, 5), L(3, 1); = J(1, 1), K(3, 0), L(0, ); = 11. J(, ), K(, ), L( 1, 0); = 1 1. J(3, 5), K(, 1), L(0, 3); = 3 In ercises 13 16, graph the polgon and its image after a reflection in the given line. (See amples and 3.) 13. = 1. = = 16. = hapter Transformations

18 In ercises 17 0, graph RST with vertices R(, 1), S(7, 3), and T(6, ) and its image after the glide reflection. (See ample.) 17. Translation: (, ) (, 1) Reflection: in the -ais 7. MOLING WITH MTHMTIS You park at some point K on line n. You deliver a pizza to House H, go back to our car, and deliver a pizza to House J. ssuming that ou can cut across both lawns, how can ou determine the parking location K that minimizes the distance HK + KJ? (See ample 6.) 18. Translation: (, ) ( 3, ) Reflection: in the line = Translation: (, ) (, + ) J H Reflection: in the line = 3 n 0. Translation: (, ) ( +, + ) Reflection: in the line = 8. TTNING TO RISION Use the numbers and In ercises 1, determine the number of lines of smmetr for the figure. (See ample 5.) 1. smbols to create the glide reflection resulting in the image shown.. ( 1, 5) 6 (5, 6) ( 1, 1) 3. (, ) (3, ). 6 8 Translation: (, ) Reflection: in = (, ) ( ), 5. USING STRUTUR Identif the line smmetr (if an) of each word. a. LOOK b. MOM c. OX d RROR NLYSIS escribe and correct the error in In ercises 9 3, find point on the -ais so + is a minimum. 30. (, 5), (1, 3) 31. ( 8, ), ( 1, 3) ( 1, 7), (5, ) 33. MTHMTIL ONNTIONS The line = 3 + to is a glide reflection. is reflected in the line = 1. What is the equation of the image? Section. hs_geo_pe_00.indd (1, ), (6, 1) describing the transformation. Reflections 187 1/19/15 10:01 M

19 HOW O YOU S IT? Use igure. igure 35. ONSTRUTION ollow these steps to construct a reflection of in line m. Use a compass and straightedge. Step 1 raw and line m. m Step Use one compass setting to find two points that are equidistant from on line m. Use the same compass setting to find a point on the other side of m that is the same distance from these two points. Label that point as. Step 3 Repeat Step to find points and. raw. igure 1 igure 36. USING TOOLS Use a reflective device to verif our construction in ercise MTHMTIL ONNTIONS Reflect MNQ in the line =. = M igure 3 igure a. Which figure is a reflection of igure in the line = a? plain. b. Which figure is a reflection of igure in the line = b? plain. c. Which figure is a reflection of igure in the line =? plain. d. Is there a figure that represents a glide reflection? plain our reasoning. Maintaining Mathematical roficienc Use the diagram to find the angle measure. (Section 1.5) 0. m O 1. m O. m O 3. m O. m O 5. m O 6. m O 7. m O 8. m O 9. m O Q 5 N THOUGHT ROVOKING Is the composition of a translation and a reflection commutative? (In other words, do ou obtain the same image regardless of the order in which ou perform the transformations?) Justif our answer. 39. MTHMTIL ONNTIONS oint (1, ) is the image of (3, ) after a reflection in line c. Write an equation for line c. Reviewing what ou learned in previous grades and lessons O hapter Transformations

20 .3 Rotations ssential Question How can ou rotate a figure in a coordinate plane? Rotating a Triangle in a oordinate lane ONSTRUTING VIL RGUMNTS To be proficient in math, ou need to use previousl established results in constructing arguments Work with a partner. a. Use dnamic geometr software to draw an triangle and label it. b. Rotate the triangle 90 counterclockwise about the origin to form. c. What is the relationship between the coordinates of the vertices of and those of? d. What do ou observe about the side lengths and angle measures of the two triangles? Rotating a Triangle in a oordinate lane Work with a partner. a. The point (, ) is rotated 90 counterclockwise about the origin. Write a rule to determine the coordinates of the image of (, ). b. Use the rule ou wrote in part (a) to rotate 90 counterclockwise about the origin. What are the coordinates of the vertices of the image,? c. raw. re its side lengths the same as those of? Justif our answer. Rotating a Triangle in a oordinate lane Work with a partner. a. The point (, ) is rotated 180 counterclockwise about the origin. Write a rule to determine the coordinates of the image of (, ). plain how ou found the rule. b. Use the rule ou wrote in part (a) to rotate (from ploration ) 180 counterclockwise about the origin. What are the coordinates of the vertices of the image,? Sample oints (1, 3) (, 3) (, 1) (0, 0) Segments = 3 = = 3.61 ngles m = m = 90 m = ommunicate Your nswer. How can ou rotate a figure in a coordinate plane? 5. In ploration 3, rotate 180 counterclockwise about the origin. What are the coordinates of the vertices of the image,? How are these coordinates related to the coordinates of the vertices of the original triangle,? Section.3 Rotations 189

21 Lesson What You Will Learn ore Vocabular rotation, p. 190 center of rotation, p. 190 angle of rotation, p. 190 rotational smmetr, p. 193 center of smmetr, p. 193 erform rotations. erform compositions with rotations. Identif rotational smmetr. erforming Rotations ore oncept Rotations rotation is a transformation in which a figure is turned about a fied point called the center of rotation. Ras drawn from the center of rotation to a point and its image form the angle of rotation. rotation about a point through an angle of maps ever point Q in the plane to a point Q so that one of the following R R properties is true. 0 Q If Q is not the center of rotation, then Q = Q and m QQ =, or If Q is the center of rotation, then Q = Q. Q center of rotation angle of rotation irection of rotation clockwise The figure above shows a 0 counterclockwise rotation. Rotations can be clockwise or counterclockwise. In this chapter, all rotations are counterclockwise unless otherwise noted. rawing a Rotation raw a 10 rotation of about point. counterclockwise Step 1 raw a segment from to. Step raw a ra to form a 10 angle with Step 3 raw so that =. Step Repeat Steps 1 3 for each verte. raw hapter Transformations

22 USING ROTTIONS You can rotate a figure more than 360. The effect, however, is the same as rotating the figure b the angle minus 360. You can rotate a figure more than 180. The diagram shows rotations of point 130, 0, and 310 about the origin. Notice that point and its images all lie on the same circle. rotation of 360 maps a figure onto itself. You can use coordinate rules to find the coordinates of a point after a rotation of 90, 180, or 70 about the origin. ore oncept oordinate Rules for Rotations about the Origin When a point (a, b) is rotated counterclockwise about the origin, the following are true. or a rotation of 90, (a, b) ( b, a). or a rotation of 180, (a, b) ( a, b). ( b, a) (a, b) or a rotation of 70, (a, b) (b, a). ( a, b) 70 (b, a) Rotating a igure in the oordinate lane Graph quadrilateral RSTU with vertices R(3, 1), S(5, 1), T(5, 3), and U(, 1) and its image after a 70 rotation about the origin. Use the coordinate rule for a 70 rotation to find the coordinates of the vertices of the image. Then graph quadrilateral RSTU and its image. R S (a, b) (b, a) R(3, 1) R (1, 3) S(5, 1) S (1, 5) U U R T 6 T(5, 3) T ( 3, 5) U(, 1) U ( 1, ) T 6 S Monitoring rogress Help in nglish and Spanish at igideasmath.com 1. Trace and point. Then draw a 50 rotation of about point.. Graph JKL with vertices J(3, 0), K(, 3), and L(6, 0) and its image after a 90 rotation about the origin. Section.3 Rotations 191

23 erforming ompositions with Rotations ostulate ostulate.3 Rotation ostulate rotation is a rigid motion. ecause a rotation is a rigid motion, and a rigid motion preserves length and angle measure, the following statements are true for the rotation shown. =, =, = m = m, m = m, m = m ecause a rotation is a rigid motion, the omposition Theorem (Theorem.1) guarantees that compositions of rotations and other rigid motions, such as translations and reflections, are rigid motions. OMMON RROR Unless ou are told otherwise, perform the transformations in the order given. erforming a omposition Graph RS with endpoints R(1, 3) and S(, 6) and its image after the composition. Reflection: in the -ais Rotation: 90 about the origin Step 1 Graph RS. Step Reflect RS in the -ais. R S has endpoints R ( 1, 3) and S (, 6). Step 3 Rotate R S 90 about the origin. R S has endpoints R (3, 1) and S (6, ). R ( 1, 3) S (, 6) 6 R(1, 3) R (3, 1) S(, 6) 8 S (6, ) Monitoring rogress Help in nglish and Spanish at igideasmath.com 3. Graph RS from ample 3. erform the rotation first, followed b the reflection. oes the order of the transformations matter? plain.. WHT I? In ample 3, RS is reflected in the -ais and rotated 180 about the origin. Graph RS and its image after the composition. 5. Graph with endpoints (, ) and ( 1, 7) and its image after the composition. Translation: (, ) (, 1) Rotation: 90 about the origin 6. Graph TUV with vertices T(1, ), U(3, 5), and V(6, 3) and its image after the composition. Rotation: 180 about the origin Reflection: in the -ais 19 hapter Transformations

24 Identifing Rotational Smmetr figure in the plane has rotational smmetr when the figure can be mapped onto itself b a rotation of 180 or less about the center of the figure. This point is the center of smmetr. Note that the rotation can be either clockwise or counterclockwise. or eample, the figure below has rotational smmetr, because a rotation of either 90 or 180 maps the figure onto itself (although a rotation of 5 does not) The figure above also has point smmetr, which is 180 rotational smmetr. Identifing Rotational Smmetr oes the figure have rotational smmetr? If so, describe an rotations that map the figure onto itself. a. parallelogram b. regular octagon c. trapezoid a. The parallelogram has rotational smmetr. The center is the intersection of the diagonals. 180 rotation about the center maps the parallelogram onto itself. b. The regular octagon has rotational smmetr. The center is the intersection of the diagonals. Rotations of 5, 90, 135, or 180 about the center all map the octagon onto itself. c. The trapezoid does not have rotational smmetr because no rotation of 180 or less maps the trapezoid onto itself. Monitoring rogress Help in nglish and Spanish at igideasmath.com etermine whether the figure has rotational smmetr. If so, describe an rotations that map the figure onto itself. 7. rhombus 8. octagon 9. right triangle Section.3 Rotations 193

25 .3 ercises namic Solutions available at igideasmath.com Vocabular and ore oncept heck 1. OMLT TH SNTN When a point (a, b) is rotated counterclockwise about the origin, (a, b) (b, a) is the result of a rotation of.. IRNT WORS, SM QUSTION Which is different? ind both answers. What are the coordinates of the vertices of the image after a 90 counterclockwise rotation about the origin? What are the coordinates of the vertices of the image after a 70 clockwise rotation about the origin? What are the coordinates of the vertices of the image after turning the figure 90 to the left about the origin? What are the coordinates of the vertices of the image after a 70 counterclockwise rotation about the origin? Monitoring rogress and Modeling with Mathematics In ercises 3 6, trace the polgon and point. Then draw a rotation of the polgon about point using the given number of degrees. (See ample 1.) G G R J M K 6 L R 6 Q S T J 19 hapter Transformations In ercises 7 10, graph the polgon and its image after a rotation of the given number of degrees about the origin. (See ample.) Q In ercises 11 1, graph XY with endpoints X( 3, 1) and Y(, 5) and its image after the composition. (See ample 3.) 11. Translation: (, ) (, + ) Rotation: 90 about the origin 1. Rotation: 180 about the origin Translation: (, ) ( 1, + 1) 13. Rotation: 70 about the origin Reflection: in the -ais 1. Reflection: in the line = Rotation: 180 about the origin

26 In ercises 15 and 16, graph LMN with vertices L(1, 6), M(, ), and N(3, ) and its image after the composition. (See ample 3.) 15. Rotation: 90 about the origin Translation: (, ) ( 3, + ) 16. Reflection: in the -ais Rotation: 70 about the origin 6. ( 1, 1) (1, 1) (, 3) (3, ) 7. ONSTRUTION ollow these steps to construct a rotation of b angle around a point O. Use a compass and straightedge. In ercises 17 0, determine whether the figure has rotational smmetr. If so, describe an rotations that map the figure onto itself. (See ample.) O RT RSONING In ercises 1, select the angles of rotational smmetr for the regular polgon. Select all that appl G 1 H Step 1 raw,, and O, the center of rotation. Step raw O. Use the construction for coping an angle to cop at O, as shown. Then use distance O and center O to find. Step 3 Repeat Step to find points and. raw. 8. RSONING You enter the revolving door at a hotel. a. You rotate the door 180. What does this mean in the contet of the situation? plain. b. You rotate the door 360. What does this mean in the contet of the situation? plain. 9. MTHMTIL ONNTIONS Use the graph of = RROR NLYSIS In ercises 5 and 6, the endpoints of are ( 1, 1) and (, 3). escribe and correct the error in finding the coordinates of the vertices of the image after a rotation of 70 about the origin. 5. ( 1, 1) ( 1, 1) (, 3) (, 3) a. Rotate the line 90, 180, 70, and 360 about the origin. Write the equation of the line for each image. escribe the relationship between the equation of the preimage and the equation of each image. b. o ou think that the relationships ou described in part (a) are true for an line that is not vertical or horizontal? plain our reasoning. 30. MKING N RGUMNT Your friend claims that rotating a figure b 180 is the same as reflecting a figure in the -ais and then reflecting it in the -ais. Is our friend correct? plain our reasoning. Section.3 Rotations 195

27 31. RWING ONLUSIONS figure onl has point smmetr. How man rotations that map the figure onto itself can be performed before it is back where it started? 38. HOW O YOU S IT? You are finishing the puzzle. The remaining two pieces both have rotational smmetr. 3. NLYZING RLTIONSHIS Is it possible for a figure to have 90 rotational smmetr but not 180 rotational smmetr? plain our reasoning. 33. NLYZING RLTIONSHIS Is it possible for a figure to have 180 rotational smmetr but not 90 rotational smmetr? plain our reasoning THOUGHT ROVOKING an rotations of 90, 180, 70, and 360 be written as the composition of two reflections? Justif our answer. 35. USING N QUTION Inside a kaleidoscope, two mirrors are placed net to each other to form a V. The angle between the mirrors determines the number of lines of smmetr in the mirror image. Use the formula 1 n(m 1) = 180 to find the measure of 1, the angle between the mirrors, for the black glass number n of lines of smmetr. a. b. a. escribe the rotational smmetr of iece 1 and of iece. b. You pick up iece 1. How man different was can it fit in the puzzle? c. efore putting iece 1 into the puzzle, ou connect it to iece. Now how man was can it fit in the puzzle? plain. 39. USING STRUTUR polar coordinate sstem locates a point in a plane b its distance from the origin O and b the measure of an angle with its verte at the origin. or eample, the point (, 30 ) is units from the origin and m XO = 30. What are the polar coordinates of the image of point after a 90 rotation? a 180 rotation? a 70 rotation? plain RSONING Use the coordinate rules for counterclockwise rotations about the origin to write coordinate rules for clockwise rotations of 90, 180, or 70 about the origin O 1 30 X USING STRUTUR XYZ has vertices X(, 5), Y(3, 1), and Z(0, ). Rotate XYZ 90 about the point (, 1) Maintaining Mathematical roficienc The figures are congruent. Name the corresponding angles and the corresponding sides. (Skills Review Handbook) 0. T S Q R Z V Y X W Reviewing what ou learned in previous grades and lessons 1. J M K L 196 hapter Transformations

28 .1.3 What id You Learn? ore Vocabular vector, p. 17 initial point, p. 17 terminal point, p. 17 horizontal component, p. 17 vertical component, p. 17 component form, p. 17 transformation, p. 17 image, p. 17 preimage, p. 17 translation, p. 17 rigid motion, p. 176 composition of transformations, p. 176 reflection, p. 18 line of reflection, p. 18 glide reflection, p. 18 line smmetr, p. 185 line of smmetr, p. 185 rotation, p. 190 center of rotation, p. 190 angle of rotation, p. 190 rotational smmetr, p. 193 center of smmetr, p. 193 ore oncepts Section.1 Vectors, p. 17 Translations, p. 17 Section. Reflections, p. 18 oordinate Rules for Reflections, p. 183 Section.3 Rotations, p. 190 oordinate Rules for Rotations about the Origin, p. 191 ostulate.1 Translation ostulate, p. 176 Theorem.1 omposition Theorem, p. 176 ostulate. Reflection ostulate, p. 18 Line Smmetr, p. 185 ostulate.3 Rotation ostulate, p. 19 Rotational Smmetr, p. 193 Mathematical ractices 1. How could ou determine whether our results make sense in ercise 6 on page 179?. State the meaning of the numbers and smbols ou chose in ercise 8 on page escribe the steps ou would take to arrive at the answer to ercise 9 part (a) on page 195. Stud Skills Keeping a ositive ttitude ver feel frustrated or overwhelmed b math? You re not alone. Just take a deep breath and assess the situation. Tr to find a productive stud environment, review our notes and eamples in the tetbook, and ask our teacher or peers for help. 197

29 .1.3 Quiz Graph quadrilateral with vertices (, 1), ( 3, 3), (0, 1), and (, 0) and its image after the translation. (Section.1) 1. (, ) ( +, ). (, ) ( 1, 5) 3. (, ) ( + 3, + 6) Graph the polgon with the given vertices and its image after a reflection in the given line. (Section.). ( 5, 6), ( 7, 8), ( 3, 11); -ais 5. ( 5, 1), (, 1), ( 1, 3); = 6. J( 1, ), K(, 5), L(5, ), M(, 1); = 3 7. (, ), Q(6, 1), R(9, ), S(6, 6); = Graph with vertices (, 1), (5, ), and (8, ) and its image after the glide reflection. (Section.) 8. Translation: (, ) (, + 6) 9. Translation: (, ) ( 9, ) Reflection: in the -ais Reflection: in the line = 1 etermine the number of lines of smmetr for the figure. (Section.) Graph the polgon and its image after a rotation of the given number of degrees about the origin. (Section.3) G H I J K Graph LMN with vertices L( 3, ), M( 1, 1), and N(, 3) and its image after the composition. (Sections.1.3) 17. Translation: (, ) (, + 3) Rotation: 180 about the origin 18. Rotation: 90 about the origin Reflection: in the -ais 19. The figure shows a game in which the object is to create solid rows using the pieces given. Using onl translations and rotations, describe the transformations for each piece at the top that will form two solid rows at the bottom. (Section.1 and Section.3) 198 hapter Transformations

30 . ongruence and Transformations ssential Question What conjectures can ou make about a figure reflected in two lines? Reflections in arallel Lines ONSTRUTING VIL RGUMNTS To be proficient in math, ou need to make conjectures and justif our conclusions. Work with a partner. Use dnamic geometr software to draw an scalene triangle and label it. a. raw an line. Reflect Sample in to form. b. raw a line parallel to. Reflect in the new line to form. c. raw the line through point that is perpendicular to. What do ou notice? d. ind the distance between points and. ind the distance between the two parallel lines. What do ou notice? e. Hide. Is there a single transformation that maps to? plain. f. Make conjectures based on our answers in parts (c) (e). Test our conjectures b changing and the parallel lines. Reflections in Intersecting Lines Work with a partner. Use dnamic geometr software to draw an scalene triangle and label it. a. raw an line. Reflect Sample in to form. b. raw an line so that angle is less than or equal to 90. Reflect in to form. c. ind the measure of. Rotate counterclockwise about point using an angle twice the measure of. d. Make a conjecture about a figure reflected in two intersecting lines. Test our conjecture b changing and the lines. ommunicate Your nswer 3. What conjectures can ou make about a figure reflected in two lines?. oint Q is reflected in two parallel lines, GH and JK, to form Q and Q. The distance from GH to JK is 3. inches. What is the distance QQ? Section. ongruence and Transformations 199

31 . Lesson What You Will Learn ore Vocabular congruent figures, p. 00 congruence transformation, p. 01 Identif congruent figures. escribe congruence transformations. Use theorems about congruence transformations. Identifing ongruent igures Two geometric figures are congruent figures if and onl if there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other. ongruent figures have the same size and shape. ongruent Not congruent same size and shape different sizes or shapes You can identif congruent figures in the coordinate plane b identifing the rigid motion or composition of rigid motions that maps one of the figures onto the other. Recall from ostulates.1.3 and Theorem.1 that translations, reflections, rotations, and compositions of these transformations are rigid motions. Identifing ongruent igures Identif an congruent figures in the coordinate plane. plain. I H 5 Square NQR is a translation of square units left and 6 units down. So, square and square NQR are congruent. J G M K Q 5 KLM is a reflection of G in the -ais. So, G and KLM are congruent. L R N U STU is a 180 rotation of HIJ. So, HIJ and STU are congruent. 5 S T Monitoring rogress Help in nglish and Spanish at igideasmath.com 1. Identif an congruent figures in the coordinate plane. plain. I J H G L T Q M K S U R N 00 hapter Transformations

32 RING You can read the notation as parallelogram,,,. ongruence Transformations nother name for a rigid motion or a combination of rigid motions is a congruence transformation because the preimage and image are congruent. The terms rigid motion and congruence transformation are interchangeable. escribe a congruence transformation that maps to GH. escribing a ongruence Transformation G H The two vertical sides of rise from left to right, and the two vertical sides of GH fall from left to right. If ou reflect in the -ais, as shown, then the image,, will have the same orientation as GH. Then ou can map to GH using a translation of units down. G H So, a congruence transformation that maps to GH is a reflection in the -ais followed b a translation of units down. Monitoring rogress Help in nglish and Spanish at igideasmath.com. In ample, describe another congruence transformation that maps to GH. 3. escribe a congruence transformation that maps JKL to MN. K L J M N Section. ongruence and Transformations 01

33 Using Theorems about ongruence Transformations ompositions of two reflections result in either a translation or a rotation. composition of two reflections in parallel lines results in a translation, as described in the following theorem. Theorem Theorem. Reflections in arallel Lines Theorem If lines k and m are parallel, then a reflection in line k followed b a reflection in line m is the same as a translation. If is the image of, then 1. is perpendicular to k and m, and. = d, where d is the distance between k and m. roof. 31, p. 06 k d m Using the Reflections in arallel Lines Theorem In the diagram, a reflection in line k maps GH to G H. reflection in line m maps G H to G H. lso, H = 9 and H =. a. Name an segments congruent to each segment: GH, H, and G. b. oes =? plain. c. What is the length of GG? H G k G H H G m a. GH G H, and GH G H. H H. G G. b. Yes, = because GG and HH are perpendicular to both k and m. So, and are opposite sides of a rectangle. c. the properties of reflections, H = 9 and H =. The Reflections in arallel Lines Theorem implies that GG = HH =, so the length of GG is (9 + ) = 6 units. Monitoring rogress Help in nglish and Spanish at igideasmath.com Use the figure. The distance between line k and line m is 1.6 centimeters.. The preimage is reflected in line k, then in line m. escribe a single transformation that maps the blue figure to the green figure. 5. What is the relationship between and line k? plain. k m 6. What is the distance between and? 0 hapter Transformations

34 composition of two reflections in intersecting lines results in a rotation, as described in the following theorem. Theorem Theorem.3 Reflections in Intersecting Lines Theorem If lines k and m intersect at point, then a reflection in line k followed b a reflection in line m is the same as a rotation about point. The angle of rotation is, where is the measure of the acute or right angle formed b lines k and m. m k roof. 31, p. 50 m = Using the Reflections in Intersecting Lines Theorem In the diagram, the figure is reflected in line k. The image is then reflected in line m. escribe a single transformation that maps to. m 70 k the Reflections in Intersecting Lines Theorem, a reflection in line k followed b a reflection in line m is the same as a rotation about point. The measure of the acute angle formed between lines k and m is 70. So, b the Reflections in Intersecting Lines Theorem, the angle of rotation is (70 ) = 10. single transformation that maps to is a 10 rotation about point. You can check that this is correct b tracing lines k and m and point, then rotating the point 10. Monitoring rogress 7. In the diagram, the preimage is reflected in line k, then in line m. escribe a single transformation that maps the blue figure onto the green figure. 8. rotation of 76 maps to. To map to using two reflections, what is the measure of the angle formed b the intersecting lines of reflection? Help in nglish and Spanish at igideasmath.com 80 m k Section. ongruence and Transformations 03

35 . ercises namic Solutions available at igideasmath.com Vocabular and ore oncept heck 1. OMLT TH SNTN Two geometric figures are if and onl if there is a rigid motion or a composition of rigid motions that moves one of the figures onto the other.. VOULRY Wh is the term congruence transformation used to refer to a rigid motion? Monitoring rogress and Modeling with Mathematics In ercises 3 and, identif an congruent figures in the coordinate plane. plain. (See ample 1.) 3.. J H M L K N U G T V 3 N M S T J U Q S V R Q R K 6. 6 W Z X Y R Q 6 S In ercises 7 10, determine whether the polgons with the given vertices are congruent. Use transformations to eplain our reasoning. 7. Q(, ), R(5, ), S(, 1) and T(6, ), U(9, ), V(8, 1) 8. W( 3, 1), X(, 1), Y(, ), Z( 5, ) and ( 1, 3), ( 1, ), (, ), (, 5) 9. J(1, 1), K(3, ), L(, 1) and M(6, 1), N(5, ), (, 1) G H L 10. (0, 0), (1, ), (, ), (3, 0) and (0, 5), ( 1, 3), G(, 3), H( 3, 5) In ercises 5 and 6, describe a congruence transformation that maps the blue preimage to the green image. (See ample.) 5. 6 G In ercises 11 1, k m, is reflected in line k, and is reflected in line m. (See ample 3.) 11. translation maps onto which triangle? 1. Which lines are perpendicular to? 13. If the distance between k and m is.6 inches, what is the length of? 1. Is the distance from to m the same as the distance from to m? plain. k m 0 hapter Transformations

36 In ercises 15 and 16, find the angle of rotation that maps onto. (See ample.) 15. m In ercises 19, find the measure of the acute or right angle formed b intersecting lines so that can be mapped to using two reflections. 19. rotation of 8 maps to. 0. rotation of maps to. 55 k 1. The rotation (, ) (, ) maps to.. The rotation (, ) (, ) maps to m k 3. RSONING Use the Reflections in arallel Lines Theorem (Theorem.) to eplain how ou can make a glide reflection using three reflections. How are the lines of reflection related?. RWING ONLUSIONS The pattern shown is called a tessellation. 17. RROR NLYSIS escribe and correct the error in describing the congruence transformation. a. What transformations did the artist use when creating this tessellation? b. re the individual figures in the tessellation congruent? plain our reasoning. is mapped to b a translation 3 units down and a reflection in the -ais. 18. RROR NLYSIS escribe and correct the error in using the Reflections in Intersecting Lines Theorem (Theorem.3). 7 RITIL THINKING In ercises 5 8, tell whether the statement is alwas, sometimes, or never true. plain our reasoning. 5. congruence transformation changes the size of a figure. 6. If two figures are congruent, then there is a rigid motion or a composition of rigid motions that maps one figure onto the other. 7. The composition of two reflections results in the same image as a rotation. 8. translation results in the same image as the composition of two reflections. 7 rotation about point maps the blue image to the green image. 9. RSONING uring a presentation, a marketing representative uses a projector so everone in the auditorium can view the advertisement. Is this projection a congruence transformation? plain our reasoning. Section. ongruence and Transformations 05

37 30. HOW O YOU S IT? What tpe of congruence transformation can be used to verif each statement about the stained glass window? a. Triangle 5 is congruent to Triangle 8. b. Triangle 1 is congruent to Triangle. c. Triangle is congruent to Triangle 7. d. entagon 3 is congruent to entagon ROVING THORM rove the Reflections in arallel Lines Theorem (Theorem.). K J J d 7 m K K Given reflection in line maps JK to J K, a reflection in line m maps J K to J K, and m. rove a. KK is perpendicular to and m. b. KK = d, where d is the distance between and m. J 33. MKING N RGUMNT Q, with endpoints (1, 3) and Q(3, ), is reflected in the -ais. The image Q is then reflected in the -ais to produce the image Q. One classmate sas that Q is mapped to Q b the translation (, ) (, 5). nother classmate sas that Q is mapped to Q b a ( 90), or 180, rotation about the origin. Which classmate is correct? plain our reasoning. 3. RITIL THINKING oes the order of reflections for a composition of two reflections in parallel lines matter? or eample, is reflecting XYZ in line and then its image in line m the same as reflecting XYZ in line m and then its image in line? Y X Z ONSTRUTION In ercises 35 and 36, cop the figure. Then use a compass and straightedge to construct two lines of reflection that produce a composition of reflections resulting in the same image as the given transformation. 35. Translation: 36. Rotation about : XYZ X Y Z m 3. THOUGHT ROVOKING tessellation is the covering of a plane with congruent figures so that there are no gaps or overlaps (see ercise ). raw a tessellation that involves two or more tpes of transformations. escribe the transformations that are used to create the tessellation. Z X Y Y Z X Maintaining Mathematical roficienc Solve the equation. heck our solution. (Skills Review Handbook) Reviewing what ou learned in previous grades and lessons = m = m 39. b + 8 = 6b 0. 7w 9 = 13 w 1. 7(n + 11) = n. (8 ) = 6 3. Last ear, the track team s ard sale earned $500. This ear, the ard sale earned $65. What is the percent of increase? (Skills Review Handbook) 06 hapter Transformations

38 .5 ilations ssential Question What does it mean to dilate a figure? ilating a Triangle in a oordinate lane LOOKING OR STRUTUR To be proficient in math, ou need to look closel to discern a pattern or structure. Work with a partner. Use dnamic geometr software to draw an triangle and label it. a. ilate using a scale factor of and a center of dilation at the origin to form. ompare the coordinates, side lengths, and angle measures of and. Sample 6 oints 5 (, 1) 3 (1, 3) (3, ) Segments =. =. = ngles m = m = m = b. Repeat part (a) using a scale factor of 1. c. What do the results of parts (a) and (b) suggest about the coordinates, side lengths, and angle measures of the image of after a dilation with a scale factor of k? ilating Lines in a oordinate lane Work with a partner. Use dnamic geometr software to draw that passes through the origin and that does not pass through the origin. a. ilate using a scale factor of 3 and a center of dilation at the origin. escribe the image. b. ilate using a scale factor 1 of 3 and a center of dilation at 0 the origin. escribe the image. 3 1 c. Repeat parts (a) and (b) using a scale factor of 1. 1 d. What do ou notice about dilations of lines passing through the center of dilation and dilations of lines not passing through the center of dilation? ommunicate Your nswer 3. What does it mean to dilate a figure? Sample. Repeat ploration 1 using a center of dilation at a point other than the origin. oints (, ) (0, 0) (, 0) Lines + = 0 + = Section.5 ilations 07

39 .5 Lesson What You Will Learn ore Vocabular dilation, p. 08 center of dilation, p. 08 scale factor, p. 08 enlargement, p. 08 reduction, p. 08 Identif and perform dilations. Solve real-life problems involving scale factors and dilations. Identifing and erforming ilations ore oncept ilations dilation is a transformation in which a figure is enlarged or reduced with respect to a fied point called the center of dilation and a scale factor k, which is the ratio of the lengths of the corresponding sides of the image and the preimage. dilation with center of dilation and scale factor k maps ever point in a figure to a point so that the following are true. If is the center point, then =. If is not the center point, then the image Q point lies on. The scale factor k is a positive number such that k =. R Q R ngle measures are preserved. dilation does not change an line that passes through the center of dilation. dilation maps a line that does not pass through the center of dilation to a parallel line. In the figure above, R R, Q Q, and QR Q R. When the scale factor k > 1, a dilation is an enlargement. When 0 < k < 1, a dilation is a reduction. Identifing ilations RING The scale factor of a dilation can be written as a fraction, decimal, or percent. ind the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. a. b a. ecause = 1 8, the scale factor is k = 3. So, the dilation is an enlargement. b. ecause = 18 30, the scale factor is k = 3. So, the dilation is a reduction. 5 Monitoring rogress Help in nglish and Spanish at igideasmath.com 1. In a dilation, = 3 and = 1. ind the scale factor. Then tell whether the dilation is a reduction or an enlargement. 08 hapter Transformations

40 RING IGRMS In this chapter, for all of the dilations in the coordinate plane, the center of dilation is the origin unless otherwise noted. ore oncept oordinate Rule for ilations If (, ) is the preimage of a point, then its image after a dilation centered at the origin (0, 0) with scale factor k is the point (k, k). ilating a igure in the oordinate lane (, ) (k, k) (, ) (k, k) Graph with vertices (, 1), (, 1), and (, 1) and its image after a dilation with a scale factor of. Use the coordinate rule for a dilation with k = to find the coordinates of the vertices of the image. Then graph and its image. (, ) (, ) (, 1) (, ) (, 1) (8, ) (, 1) (8, ) 6 Notice the relationships between the lengths and slopes of the sides of the triangles in ample. ach side length of is longer than its corresponding side b the scale factor. The corresponding sides are parallel because their slopes are the same. ilating a igure in the oordinate lane Graph quadrilateral KLMN with vertices K( 3, 6), L(0, 6), M(3, 3), and N( 3, 3) and its image after a dilation with a scale factor of 1 3. Use the coordinate rule for a dilation with k = 1 3 to find the coordinates of the vertices of the image. Then graph quadrilateral KLMN and its image. (, ) ( 1 3, 1 3 ) K( 3, 6) K ( 1, ) L(0, 6) L (0, ) M(3, 3) M (1, 1) N( 3, 3) N ( 1, 1) K N L K L M N M Monitoring rogress Help in nglish and Spanish at igideasmath.com Graph QR and its image after a dilation with scale factor k.. (, 1), Q( 1, 0), R(0, 1); k = 3. (5, 5), Q(10, 5), R(10, 5); k = 0. Section.5 ilations 09

41 onstructing a ilation Use a compass and straightedge to construct a dilation of QR with a scale factor of. Use a point outside the triangle as the center of dilation. Step 1 Step Step 3 Q Q Q Q Q R R R R R raw a triangle raw QR and choose the center of the dilation outside the triangle. raw ras from through the vertices of the triangle. Use a compass Use a compass to locate on so that = (). Locate Q and R using the same method. onnect points onnect points, Q, and R to form Q R. scale factor k center of dilation preimage In the coordinate plane, ou can have scale factors that are negative numbers. When this occurs, the figure rotates 180. So, when k > 0, a dilation with a scale factor of k is the same as the composition of a dilation with a scale factor of k followed b a rotation of 180 about the center of dilation. Using the coordinate rules for a dilation and a rotation of 180, ou can think of the notation as (, ) (k, k) ( k, k). Using a Negative Scale actor scale factor k Graph GH with vertices (, ), G(, ), and H(, ) and its image 1 after a dilation with a scale factor of. Use the coordinate rule for a dilation with k = to find the coordinates of the vertices of the image. Then graph GH and its image. (, ) ( 1, 1 ) (, ) (, 1) G(, ) G (1, ) H(, ) H (1, 1) 1 G H H G 10 hapter Transformations Monitoring rogress Help in nglish and Spanish at igideasmath.com. Graph QR with vertices (1, ), Q(3, 1), and R(1, 3) and its image after a dilation with a scale factor of. 5. Suppose a figure containing the origin is dilated. plain wh the corresponding point in the image of the figure is also the origin.

42 RING Scale factors are written so that the units in the numerator and denominator divide out. Solving Real-Life roblems inding a Scale actor You are making our own photo stickers. Your photo is inches b inches. The image on the stickers is 1.1 inches b 1.1 inches. What is the scale factor of this dilation? The scale factor is the ratio of a side length of the sticker image to a side 1.1 in. length of the original photo, or in.. So, in simplest form, the scale factor is inding the Length of an Image You are using a magnifing glass that shows the image of an object that is si times the object s actual size. etermine the length of the image of the spider seen through the magnifing glass. image length actual length = k 1.5 = 6 = in. in. 1.5 cm So, the image length through the magnifing glass is 9 centimeters. 1.6 cm Monitoring rogress Help in nglish and Spanish at igideasmath.com 6. n optometrist dilates the pupils of a patient s ees to get a better look at the back of the ees. pupil dilates from.5 millimeters to 8 millimeters. What is the scale factor of this dilation? 7. The image of a spider seen through the magnifing glass in ample 6 is shown at the left. ind the actual length of the spider. When a transformation, such as a dilation, changes the shape or size of a figure, the transformation is nonrigid. In addition to dilations, there are man possible nonrigid transformations. Two eamples are shown below. It is important to pa close attention to whether a nonrigid transformation preserves lengths and angle measures. Horizontal Stretch Vertical Stretch Section.5 ilations 11

43 .5 ercises namic Solutions available at igideasmath.com Vocabular and ore oncept heck 1. OMLT TH SNTN If (, ) is the preimage of a point, then its image after a dilation centered at the origin (0, 0) with scale factor k is the point.. WHIH ON OSN T LONG? Which scale factor does not belong with the other three? plain our reasoning. 5 60% 115% Monitoring rogress and Modeling with Mathematics In ercises 3 6, find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. (See ample 1.) ONSTRUTION In ercises 11 1, cop the diagram. Then use a compass and straightedge to construct a dilation of quadrilateral RSTU with the given center and scale factor k. R S U T 11. enter, k = 3 1. enter, k = 13. enter R, k = enter, k = 75% ONSTRUTION In ercises 7 10, cop the diagram. Then use a compass and straightedge to construct a dilation of LMN with the given center and scale factor k. L In ercises 15 18, graph the polgon and its image after a dilation with scale factor k. (See amples and 3.) 15. X(6, 1), Y(, ), Z(1, ); k = (0, 5), ( 10, 5), (5, 5); k = 10% 17. T(9, 3), U(6, 0), V(3, 9), W(0, 0); k = 3 M 18. J(, 0), K( 8, ), L(0, ), M(1, 8); k = enter, k = 8. enter, k = 3 9. enter M, k = enter, k = 5% 1 hapter Transformations N In ercises 19, graph the polgon and its image after a dilation with scale factor k. (See ample.) 19. ( 5, 10), ( 10, 15), (0, 5); k = L(0, 0), M(, 1), N( 3, 6); k = 3 1. R( 7, 1), S(, 5), T(, 3), U( 3, 3); k =. W(8, ), X(6, 0), Y( 6, ), Z(, ); k = 0.5

44 RROR NLYSIS In ercises 3 and, describe and correct the error in finding the scale factor of the dilation k = 1 3 = In ercises 31 3, ou are using a magnifing glass. Use the length of the insect and the magnification level to determine the length of the image seen through the magnifing glass. (See ample 6.) 31. emperor moth 3. ladbug Magnification: 5 Magnification: 10. k = = 1 In ercises 5 8, the red figure is the image of the blue figure after a dilation with center. ind the scale factor of the dilation. Then find the value of the variable (, ) (, 1) 1 1 n mm.5 mm 33. dragonfl 3. carpenter ant Magnification: 0 Magnification: 15 7 mm 1 mm 35. NLYZING RLTIONSHIS Use the given actual and magnified lengths to determine which of the following insects were looked at using the same magnifing glass. plain our reasoning. grasshopper black beetle ctual: in. ctual: 0.6 in. Magnified: 15 in. Magnified:. in m 7 8 honebee monarch butterfl ctual: 5 in. 8 ctual: 3.9 in. Magnified: 75 in. 16 Magnified: 9.5 in. 9. INING SL TOR You receive wallet-sized photos of our school picture. The photo is.5 inches b 3.5 inches. You decide to dilate the photo to 5 inches b 7 inches at the store. What is the scale factor of this dilation? (See ample 5.) 30. INING SL TOR Your visuall impaired friend asked ou to enlarge our notes from class so he can stud. You took notes on 8.5-inch b 11-inch paper. The enlarged cop has a smaller side with a length of 10 inches. What is the scale factor of this dilation? (See ample 5.) 36. THOUGHT ROVOKING raw and so that is a dilation of. ind the center of dilation and eplain how ou found it. 37. RSONING Your friend prints a -inch b 6-inch photo for ou from the school dance. ll ou have is an 8-inch b 10-inch frame. an ou dilate the photo to fit the frame? plain our reasoning. Section.5 ilations 13

45 38. HOW O YOU S IT? oint is the center of dilation of the images. The scale factor is 1 3. Which figure is the original figure? Which figure is the dilated figure? plain our reasoning. 39. MTHMTIL ONNTIONS The larger triangle is a dilation of the smaller triangle. ind the values of and. (3 3) 5. NLYZING RLTIONSHIS ilate the line through O(0, 0) and (1, ) using a scale factor of. a. What do ou notice about the lengths of O and O? b. What do ou notice about O and O? 6. NLYZING RLTIONSHIS ilate the line through (0, 1) and (1, ) using a scale factor of 1. a. What do ou notice about the lengths of and? b. What do ou notice about and? 7. TTNING TO RISION You are making a blueprint of our house. You measure the lengths of the walls of our room to be 11 feet b 1 feet. When ou draw our room on the blueprint, the lengths of the walls are 8.5 inches b 9 inches. What scale factor dilates our room to the blueprint? ( + 16) MKING N RGUMNT Your friend claims that dilating a figure b 1 is the same as dilating a figure b 1 because the original figure will not be enlarged or reduced. Is our friend correct? plain our reasoning WRITING plain wh a scale factor of is the same as 00%. In ercises 1, determine whether the dilated figure or the original figure is closer to the center of dilation. Use the given location of the center of dilation and scale factor k. 1. enter of dilation: inside the figure; k = 3. enter of dilation: inside the figure; k = 1 9. USING STRUTUR Rectangle WXYZ has vertices W( 3, 1), X( 3, 3), Y(5, 3), and Z(5, 1). a. ind the perimeter and area of the rectangle. b. ilate the rectangle using a scale factor of 3. ind the perimeter and area of the dilated rectangle. ompare with the original rectangle. What do ou notice? c. Repeat part (b) using a scale factor of 1. d. Make a conjecture for how the perimeter and area change when a figure is dilated. 50. RSONING You put a reduction of a page on the original page. plain wh there is a point that is in the same place on both pages. 3. enter of dilation: outside the figure; k = 10%. enter of dilation: outside the figure; k = 0.1 Maintaining Mathematical roficienc 51. RSONING has vertices (, ), (, 6), and (7, ). ind the coordinates of the vertices of the image after a dilation with center (, 0) and a scale factor of. Reviewing what ou learned in previous grades and lessons The vertices of are (, 1), (0, ), and ( 3, 5). ind the coordinates of the vertices of the image after the translation. (Section.1) 5. (, ) (, ) 53. (, ) ( 1, + 3) 5. (, ) ( + 3, 1) 55. (, ) (, ) 56. (, ) ( + 1, ) 57. (, ) ( 3, + 1) 1 hapter Transformations

46 TTNING TO RISION.6 To be proficient in math, ou need to use clear definitions in discussions with others and in our own reasoning. Similarit and Transformations ssential Question When a figure is translated, reflected, rotated, or dilated in the plane, is the image alwas similar to the original figure? Two figures are similar fi gures when the have the same shape but not necessaril the same size. ilations and Similarit Work with a partner. a. Use dnamic geometr software to draw an triangle and label it. b. ilate the triangle using a scale factor of 3. Is the image similar to the original triangle? Justif our answer Similar Triangles Sample oints (, 1) ( 1, 1) (1, 0) (0, 0) Segments =. =. = 3.16 ngles m = 5 m = 90 m = 5 G Rigid Motions and Similarit Work with a partner. a. Use dnamic geometr software to draw an triangle. b. op the triangle and translate it 3 units left and units up. Is the image similar to the original triangle? Justif our answer. c. Reflect the triangle in the -ais. Is the image similar to the original triangle? Justif our answer. d. Rotate the original triangle 90 counterclockwise about the origin. Is the image similar to the original triangle? Justif our answer. ommunicate Your nswer 3. When a figure is translated, reflected, rotated, or dilated in the plane, is the image alwas similar to the original figure? plain our reasoning.. figure undergoes a composition of transformations, which includes translations, reflections, rotations, and dilations. Is the image similar to the original figure? plain our reasoning. Section.6 Similarit and Transformations 15

47 .6 Lesson What You Will Learn ore Vocabular similarit transformation, p. 16 similar figures, p. 16 erform similarit transformations. escribe similarit transformations. rove that figures are similar. erforming Similarit Transformations dilation is a transformation that preserves shape but not size. So, a dilation is a nonrigid motion. similarit transformation is a dilation or a composition of rigid motions and dilations. Two geometric figures are similar figures if and onl if there is a similarit transformation that maps one of the figures onto the other. Similar figures have the same shape but not necessaril the same size. ongruence transformations preserve length and angle measure. When the scale factor of the dilation(s) is not equal to 1 or 1, similarit transformations preserve angle measure onl. erforming a Similarit Transformation Graph with vertices (, 1), (, ), and (, 1) and its image after the similarit transformation. Translation: (, ) ( + 5, + 1) ilation: (, ) (, ) Step 1 Graph. 8 6 (, ) (6, 6) (, 1) (3, 3) (6, ) (, ) (1, ) (3, ) (, 1) 6 8 Step Translate 5 units right and 1 unit up. has vertices (1, ), (3, 3), and (3, ). Step 3 ilate using a scale factor of. has vertices (, ), (6, 6), and (6, ). Monitoring rogress Help in nglish and Spanish at igideasmath.com 1. Graph with endpoints (, ) and (, ) and its image after the similarit transformation. Rotation: 90 about the origin ilation: (, ) ( 1, 1 ). Graph GH with vertices (1, ), G(, ), and H(, 0) and its image after the similarit transformation. Reflection: in the -ais ilation: (, ) (1.5, 1.5) 16 hapter Transformations

48 escribing Similarit Transformations escribing a Similarit Transformation escribe a similarit transformation that maps trapezoid QRS to trapezoid WXYZ. Q X W Y Z 6 S R QR falls from left to right, and XY rises from left to right. If ou reflect trapezoid QRS in the -ais as shown, then the image, trapezoid Q R S, will have the same orientation as trapezoid WXYZ. ( 6, 3) Q( 3, 3) Q (3, 3) X (6, 3) W Y Z S( 6, 3) R(0, 3) R (0, 3) S (6, 3) Trapezoid WXYZ appears to be about one-third as large as trapezoid Q R S. ilate trapezoid Q R S using a scale factor of 1 3. (, ) ( 1 3, 1 3 ) (6, 3) (, 1) Q (3, 3) Q (1, 1) R (0, 3) R (0, 1) S (6, 3) S (, 1) The vertices of trapezoid Q R S match the vertices of trapezoid WXYZ. So, a similarit transformation that maps trapezoid QRS to trapezoid WXYZ is a reflection in the -ais followed b a dilation with a scale factor of 1 3. Monitoring rogress 3. In ample, describe another similarit transformation that maps trapezoid QRS to trapezoid WXYZ.. escribe a similarit transformation that maps quadrilateral G to quadrilateral STUV. Help in nglish and Spanish at igideasmath.com U G V S T Section.6 Similarit and Transformations 17

49 roving igures re Similar To prove that two figures are similar, ou must prove that a similarit transformation maps one of the figures onto the other. roving That Two Squares re Similar rove that square is similar to square GH. Given Square with side length r, G square GH with side length s, H s r rove Square is similar to square GH. H Translate square so that point maps to point. ecause translations map segments to parallel segments and H, the image of lies on H. r G s r G s H ecause translations preserve length and angle measure, the image of,, is a square with side length r. ecause all the interior angles of a square are right angles, H. When coincides with H, coincides with. So, lies on. Net, dilate square using center of dilation. hoose the scale factor to be the ratio of the side lengths of GH and, which is s r. r G s H This dilation maps to H and to because the images of and have side length s r (r) = s and the segments and lie on lines passing through the center of dilation. So, the dilation maps to and to H. The image of lies s r (r) = s units to the right of the image of and s (r) = s units above the image of. r So, the image of is G. G s H H N similarit transformation maps square to square GH. So, square is similar to square GH. v Monitoring rogress Help in nglish and Spanish at igideasmath.com t K M 5. rove that JKL is similar to MN. Given Right isosceles JKL with leg length t, right isosceles MN with leg length v, LJ M L J rove JKL is similar to MN. 18 hapter Transformations

50 .6 ercises namic Solutions available at igideasmath.com Vocabular and ore oncept heck 1. VOULRY What is the difference between similar fi gures and congruent fi gures?. OMLT TH SNTN transformation that produces a similar figure, such as a dilation, is called a. Monitoring rogress and Modeling with Mathematics In ercises 3 6, graph GH with vertices (, ), G(, ), and H(, ) and its image after the similarit transformation. (See ample 1.) 3. Translation: (, ) ( + 3, + 1) ilation: (, ) (, ). ilation: (, ) ( 1, 1 ) Reflection: in the -ais 5. Rotation: 90 about the origin ilation: (, ) (3, 3) 6. ilation: (, ) ( 3, 3 ) Reflection: in the -ais In ercises 7 and 8, describe a similarit transformation that maps the blue preimage to the green image. (See ample.) 7. 6 T V U In ercises 9 1, determine whether the polgons with the given vertices are similar. Use transformations to eplain our reasoning. 9. (6, 0), (9, 6), (1, 6) and (0, 3), (1, 5), (, 5) 10. Q( 1, 0), R(, ), S(1, 3), T(, 1) and W(0, ), X(, ), Y(6, ), Z(, ) 11. G(, 3), H(, 3), I(, 0) and J(1, 0), K(6, ), L(1, ) 1. (, 3), (, 3), ( 1, 1), G(, 1) and L(1, 1), M(3, 1), N(6, 3), (1, 3) In ercises 13 and 1, prove that the figures are similar. (See ample 3.) 13. Given Right isosceles with leg length j, right isosceles RST with leg length k, RT rove is similar to RST. j k S R T 8. L 1. Given Rectangle JKLM with side lengths and, rectangle QRST with side lengths and rove Rectangle JKLM is similar to rectangle QRST. Q 6 R K J K Q R M S J 6 M L T S Section.6 Similarit and Transformations 19

51 15. MOLING WITH MTHMTIS etermine whether 19. NLYZING RLTIONSHIS Graph a polgon in the regular-sized stop sign and the stop sign sticker are similar. Use transformations to eplain our reasoning. a coordinate plane. Use a similarit transformation involving a dilation (where k is a whole number) and a translation to graph a second polgon. Then describe a similarit transformation that maps the second polgon onto the first. 1.6 in. in. 0. THOUGHT ROVOKING Is the composition of a rotation and a dilation commutative? (In other words, do ou obtain the same image regardless of the order in which ou perform the transformations?) Justif our answer. 16. RROR NLYSIS escribe and correct the error in comparing the figures MTHMTIL ONNTIONS Quadrilateral JKLM is mapped to quadrilateral J K L M using the dilation (, ) 3, 3. Then quadrilateral J K L M is mapped to quadrilateral J K L M using the translation (, ) ( + 3, ). The vertices of quadrilateral J K L M are J ( 1, 0), K ( 1, 18), L ( 6, 18), and M ( 6, 0). ind the coordinates of the vertices of quadrilateral JKLM and quadrilateral J K L M. re quadrilateral JKLM and quadrilateral J K L M similar? plain ) ( 1 1 igure is similar to igure. 17. MKING N RGUMNT member of the homecoming decorating committee gives a printing compan a banner that is 3 inches b 1 inches to enlarge. The committee member claims the banner she receives is distorted. o ou think the printing compan distorted the image she gave it? plain.. RT RSONING Use the diagram. 6 R 8 in. S Q 18 in. of figures is similar. plain our reasoning. b. lassif the angle as acute, obtuse, right, or straight.. hapter hs_geo_pe_006.indd 0 Reviewing what ou learned in previous grades and lessons (Section 1.5) b. Repeat part (a) for two other triangles. What conjecture can ou make? Maintaining Mathematical roficienc 3. 6 a. onnect the midpoints of the sides of QRS to make another triangle. Is this triangle similar to QRS? Use transformations to support our answer. 18. HOW O YOU S IT? etermine whether each pair a. Transformations 3/9/16 9:1 M

52 ..6 What id You Learn? ore Vocabular congruent figures, p. 00 congruence transformation, p. 01 dilation, p. 08 center of dilation, p. 08 scale factor, p. 08 enlargement, p. 08 reduction, p. 08 similarit transformation, p. 16 similar figures, p. 16 ore oncepts Section. Identifing ongruent igures, p. 00 escribing a ongruence Transformation, p. 01 Theorem. Reflections in arallel Lines Theorem, p. 0 Theorem.3 Reflections in Intersecting Lines Theorem, p. 03 Section.5 ilations and Scale actor, p. 08 oordinate Rule for ilations, p. 09 Negative Scale actors, p. 10 Section.6 Similarit Transformations, p. 16 Mathematical ractices 1. Revisit ercise 31 on page 06. Tr to recall the process ou used to reach the solution. id ou have to change course at all? If so, how did ou approach the situation?. escribe a real-life situation that can be modeled b ercise 8 on page 13. erformance Task The Magic of Optics Look at ourself in a shin spoon. What happened to our reflection? an ou describe this mathematicall? Now turn the spoon over and look at our reflection on the back of the spoon. What happened? Wh? To eplore the answers to these questions and more, go to igideasmath.com. 1

53 hapter Review.1 Translations (pp ) namic Solutions available at igideasmath.com Graph quadrilateral with vertices (1, ), (3, 1), (0, 3), and (, 1) and its image after the translation (, ) ( +, ). Graph quadrilateral. To find the coordinates of the vertices of the image, add to the -coordinates and subtract from the -coordinates of the vertices of the preimage. Then graph the image. (, ) ( +, ) (1, ) (3, ) (3, 1) (5, 3) (0, 3) (, 1) (, 1) (, 1) Graph XYZ with vertices X(, 3), Y( 3, ), and Z(, 3) and its image after the translation. 1. (, ) (, + ). (, ) ( 3, ) 3. (, ) ( + 3, 1). (, ) ( +, + 1) Graph QR with vertices (0, ), Q(1, 3), and R(, 5) and its image after the composition. 5. Translation: (, ) ( + 1, + ) 6. Translation: (, ) (, + 3) Translation: (, ) (, + 1) Translation: (, ) ( 1, + 1). Reflections (pp ) Graph with vertices (1, 1), (3, ), and (, ) and its image after a reflection in the line =. Graph and the line =. Then use the coordinate rule for reflecting in the line = to find the coordinates of the vertices of the image. (a, b) (b, a) (1, 1) ( 1, 1) (3, ) (, 3) (, ) (, ) Graph the polgon and its image after a reflection in the given line. 7. = 6 8. = 3 H = G 6 9. How man lines of smmetr does the figure have? hapter Transformations

54 .3 Rotations (pp ) Graph LMN with vertices L(1, 1), M(, 3), and N(, 0) and its image after a 70 rotation about the origin. Use the coordinate rule for a 70 rotation to find the coordinates of the vertices of the image. Then graph LMN and its image. M L(1, 1) L ( 1, 1) M(, 3) M (3, ) (a, b) (b, a) L L N M N(, 0) N (0, ) Graph the polgon with the given vertices and its image after a rotation of the given number of degrees about the origin. 10. ( 3, 1), (, ), (3, 3); W(, 1), X( 1, 3), Y(3, 3), Z(3, 3); Graph XY with endpoints X(5, ) and Y(3, 3) and its image after a reflection in the -ais and then a rotation of 70 about the origin. etermine whether the figure has rotational smmetr. If so, describe an rotations that map the figure onto itself N. ongruence and Transformations (pp ) escribe a congruence transformation that maps quadrilateral to quadrilateral WXYZ, as shown at the right. falls from left to right, and WX rises from left to right. If ou reflect quadrilateral in the -ais as shown at the bottom right, then the image, quadrilateral, will have the same orientation as quadrilateral WXYZ. Then ou can map quadrilateral to quadrilateral WXYZ using a translation of 5 units left. So, a congruence transformation that maps quadrilateral to quadrilateral WXYZ is a reflection in the -ais followed b a translation of 5 units left. escribe a congruence transformation that maps to JKL. 15. (, 1), (, 1), (1, ) and J(, ), K(, ), L( 1, 1) 16. ( 3, ), ( 5, 1), ( 1, 1) and J(1, ), K( 1, 1), L(3, 1) Y Z Y Z X W X 17. Which transformation is the same as reflecting an object in two parallel lines? in two intersecting lines? W hapter hapter Review 3

55 .5 ilations (pp. 07 1) Graph trapezoid with vertices (1, 1), (1, 3), (3, ), and (3, 1) and its image after a dilation with a scale factor of. Use the coordinate rule for a dilation with k = to find the coordinates of the vertices of the image. Then graph trapezoid and its image. 6 (, ) (, ) (1, 1) (, ) (1, 3) (, 6) (3, ) (6, ) 6 (3, 1) (6, ) Graph the triangle and its image after a dilation with scale factor k. 18. (, ), Q(, ), R(8, ); k = X( 3, ), Y(, 3), Z(1, 1); k = 3 0. You are using a magnifing glass that shows the image of an object that is eight times the object s actual size. The image length is 15. centimeters. ind the actual length of the object..6 Similarit and Transformations (pp. 15 0) escribe a similarit transformation that maps GH to LMN, as shown at the right. G is horizontal, and LM is vertical. If ou rotate GH 90 about the origin as shown at the bottom right, then the image, G H, will have the same orientation as LMN. LMN appears to be half as large as G H. ilate G H using a scale factor of 1. (, ) ( 1, 1 ) (, ) ( 1, 1) G (, 6) G ( 1, 3) H ( 6, ) H ( 3, ) The vertices of G H match the vertices of LMN. So, a similarit transformation that maps GH to LMN is a rotation of 90 about the origin followed b a dilation with a scale factor of 1. 6 H 6 6 M N L G 6 M N L H G 6 H G 6 escribe a similarit transformation that maps to RST. 1. (1, 0), (, 1), ( 1, ) and R( 3, 0), S(6, 3), T(3, 6). (6, ), (, 0), (, ) and R(, 3), S(0, 1), T(1, ) 3. (3, ), (0, ), ( 1, 3) and R(, 6), S(8, 0), T( 6, ) hapter Transformations

56 hapter Test Graph RST with vertices R(, 1), S(, ), and T(3, ) and its image after the translation. 1. (, ) (, + 1). (, ) ( +, ) Graph the polgon with the given vertices and its image after a rotation of the given number of degrees about the origin. 3. ( 1, 1), ( 3, ), (1, ); 70. J( 1, 1), K(3, 3), L(, 3), M(0, ); 90 etermine whether the polgons with the given vertices are congruent or similar. Use transformations to eplain our reasoning. 5. Q(, ), R(5, ), S(6, ), T(1, ) and 6. ( 6, 6), ( 6, ), (, ) and W(6, 1), X(15, 1), Y(18, 6), Z(3, 6) (9, 7), (5, 7), ( 1, 3) etermine whether the object has line smmetr and whether it has rotational smmetr. Identif all lines of smmetr and angles of rotation that map the figure onto itself raw a diagram using a coordinate plane, two parallel lines, and a parallelogram that demonstrates the Reflections in arallel Lines Theorem (Theorem.). 11. rectangle with vertices W(, ), X(, ), Y(, ), and Z(, ) is reflected in the -ais. Your friend sas that the image, rectangle W X Y Z, is eactl the same as the preimage. Is our friend correct? plain our reasoning. 1. Write a composition of transformations that maps onto in the tesselation shown. Is the composition a congruence transformation? plain our reasoning There is one slice of a large pizza and one slice of a small pizza in the bo. a. escribe a similarit transformation that maps pizza slice to pizza slice. b. What is one possible scale factor for a medium slice of pizza? plain our reasoning. (Use a dilation on the large slice of pizza.) 1. The original photograph shown is inches b 6 inches. a. What transformations can ou use to produce the new photograph? b. You dilate the original photograph b a scale factor of 1. What are the dimensions of the new photograph? c. You have a frame that holds photos that are 8.5 inches b 11 inches. an ou dilate the original photograph to fit the frame? plain our reasoning. new original hapter hapter Test 5