Properties of Transformations

Transcription

1 Lesson G.O N.VM G.O G.O G.O G.O G.SRT. 9 roperties of Transformations 9. Translate igures and Use Vectors 9.2 Use roperties of Matrices 9.3 erform Reflections 9.4 erform Rotations 9.5 ppl ompositions of Transformations 9.6 Identif Smmetr 9.7 Identif and erform Dilations efore reviousl, ou learned the following skills, which ou ll use in this chapter: translating, reflecting, and rotating polgons, and using similar polgons. rerequisite Skills VOULRY HEK Match the transformation of Triangle with its graph.. Translation of Triangle D 2. Reflection of Triangle 3. Rotation of Triangle SKILLS ND LGER HEK The vertices of JKLM are J(2, 6), K(2, 5), L(2, 2), and M(2, ). Graph its image after the transformation described. 4. Reflect in the -ais. 5. Reflect in the -ais. In the diagram, D, EGH. 6. ind the scale factor of D to EGH. 7. ind the values of,, and z. z D 0 6 E 2 H 5 8 G 562 raldo de Luca/orbis

2 Now In this chapter, ou will appl the big ideas listed below and reviewed in the hapter Summar. You will also use the ke vocabular listed below. ig Ideas erforming congruence and similarit transformations 2 Making real-world connections to smmetr and tessellations 3 ppling matrices and vectors in Geometr KEY VOULRY image preimage isometr vector component form matri element dimensions line of reflection center of rotation angle of rotation glide reflection composition of transformations line smmetr rotational smmetr scalar multiplication Wh? You can use properties of shapes to determine whether shapes tessellate. or eample, ou can use angle measurements to determine which shapes can be used to make a tessellation. Geometr The animation illustrated below helps ou answer a question from this chapter: How can ou use tiles to tessellate a floor? Rotate lockwise Rotate ounterclockwise Reflect cross -ais Reflect crosss -ais s tessellation covers a plane with no gaps or overlaps. Start ontinue hoose tiles and draw a tessellation. You ma translate, reflect, and rotate tiles. Geometr at m.hrw.com 563

3 9. Translate igures and Use Vectors efore You used a coordinate rule to translate a figure. Now You will use a vector to translate a figure. Wh? So ou can find a distance covered on snowshoes, as in Es Ke Vocabular image preimage isometr vector initial point, terminal point, horizontal component, vertical component component form translation transformation 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. Recall that a translation moves ever point of a figure the same distance in the same direction. More specificall, a translation maps, or moves, the points and Q of a plane figure to the points 9 (read prime ) and Q9, so that one of the following statements is true: QQ9 and } 9 i } QQ9, or 9 5 QQ9 and } 9 and } QQ9 are collinear. Q9 Q E XMLE Translate a figure in the coordinate plane Graph quadrilateral D with vertices (2, 2), (2, 5), (4, 6), and D(4, 2). ind the image of each verte after the translation (, ) ( 3, 2 ). Then graph the image using prime notation. USE NOTTION You can use prime notation to name an image. or eample, if the preimage is n, then its image is n 999, read as triangle prime, prime, prime. Solution irst, draw D. ind the translation of each verte b adding 3 to its -coordinate and subtracting from its -coordinate. Then graph the image. (, ) ( 3, 2 ) (2, 2) 9(2, ) 9 9 (2, 5) 9(2, 4) (4, 6) 9(7, 5) D(4, 2) D9(7, ) D 9 D9.9-2.G.O.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometr software. Specif a sequence of transformations that will carr a given figure onto another. GUIDED RTIE for Eample 564 hapter 9 roperties of Transformations. Draw n RST with vertices R(2, 2), S(5, 2), and T(3, 5). ind the image of each verte after the translation (, ) (, 2). Graph the image using prime notation. 2. The image of (, ) ( 4, 2 7) is } 9Q9 with endpoints 9(23, 4) and Q9(2, ). ind the coordinates of the endpoints of the preimage. Roalt-ree/orbis

4 ISOMETRY n isometr is a transformation that preserves length and angle measure. Isometr is another word for congruence transformation. RED DIGRMS In this book, the preimage is alwas shown in blue, and the image is alwas shown in red. E XMLE 2 Write a translation rule and verif congruence Write a rule for the translation of n to n 999. Then verif that the transformation is an isometr. Solution To go from to 9, move 4 units left and unit up. So, a rule for the translation is (, ) ( 2 4, ). Use the SS ongruence ostulate. Notice that , and The slopes of } and } 99 are 0, and the slopes of } and } 99 are undefined, so the sides are perpendicular. Therefore, and 9 are congruent right angles. So, n > n 999. The translation is an isometr GUIDED RTIE for Eample 2 3. In Eample 2, write a rule to translate n 999 back to n. THEOREM or Your Notebook THEOREM 9. Translation Theorem translation is an isometr n > n 999 ROO Translation Theorem translation is an isometr. GIVEN c (a, b) and Q(c, d) are two points on a figure translated b (, ) ( s, t). ROVE c Q 5 9Q9 (a, b) Q(c, d) 9(a s, b t) Q9(c s, d t) The translation maps (a, b) to 9(a s, b t) and Q(c, d) to Q9(c s, d t). Use the Distance ormula to find Q and 9Q9. Q 5 Ï }} (c 2 a) 2 (d 2 b) 2. 9Q95 Ï }}}} [(c s) 2 (a s)] 2 [(d t) 2 (b t)] 2 5 Ï }}}} (c s 2 a 2 s) 2 (d t 2 b 2 t) 2 5 Ï }} (c 2 a) 2 (d 2 b) 2 Therefore, Q 5 9Q9 b the Transitive ropert of Equalit. 9. Translate igures and Use Vectors 565

5 VETORS nother wa to describe a translation is b using a vector. vector is a quantit that has both direction and magnitude, or size. vector is represented in the coordinate plane b an arrow drawn from one point to another. KEY ONET or Your Notebook USE NOTTION Use brackets to write the component form of the vector r, s. Use parentheses to write the coordinates of the point ( p, q). r, s (p, q) Vectors The diagram shows a vector named G # z, read as vector G. The initial point, or starting point, of the vector is. horizontal component 5 units right G 3 units up The terminal point, or ending point, of the vector is G. vertical component The component form of a vector combines the horizontal and vertical components. So, the component form of G # z is 5, 3. E XMLE 3 Identif vector components Name the vector and write its component form. a. b. T S Solution a. The vector is # z. rom initial point to terminal point, ou move 9 units right and 2 units down. So, the component form is 9, 22. b. The vector is ST # z. rom initial point S to terminal point T, ou move 8 units left and 0 units verticall. The component form is 28, 0. E XMLE 4 Use a vector to translate a figure The vertices of n are (0, 3), (2, 4), and (, 0). Translate n using the vector 5, 2. USE VETORS Notice that the vector can have different initial points. The vector describes onl the direction and magnitude of the translation. Solution irst, graph n. Use 5, 2 to move each verte 5 units to the right and unit down. Label the image vertices. Draw n 999. Notice that the vectors drawn from preimage to image vertices are parallel. 9(5, 2) 9(7, 3) 7 9(6, 2) 566 hapter 9 roperties of Transformations

6 GUIDED RTIE for Eamples 3 and 4 Name the vector and write its component form. 4. R S 5. X T 6. K 7. The vertices of n LMN are L(2, 2), M(5, 3), and N(9, ). Translate n LMN using the vector 22, 6. E XMLE 5 Solve a multi-step problem NVIGTION boat heads out from point on one island toward point D on another. The boat encounters a storm at, 2 miles east and 4 miles north of its starting point. The storm pushes the boat off course to point, as shown. N (2, 4) D(8, 5) (6, 2) (0, 0) a. Write the component form of # z. b. Write the component form of # z. c. Write the component form of the vector that describes the straight line path from the boat s current position to its intended destination D. Solution a. The component form of the vector from (0, 0) to (2, 4) is # z , , 4. b. The component form of the vector from (2, 4) to (6, 2) is # z , , 22. c. The boat is currentl at point and needs to travel to D. The component form of the vector from (6, 2) to D(8, 5) is D # z , , 3. GUIDED RTIE for Eample 5 8. WHT I? In Eample 5, suppose there is no storm. Write the component form of the vector that describes the straight path from the boat s starting point to its final destination D. 9. Translate igures and Use Vectors 567

7 9. EXERISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 7,, and 35 5 STNDRDIZED TEST RTIE Es. 2, 4, and 42 SKILL RTIE. VOULRY op and complete:? is a quantit that has both? and magnitude. 2. WRITING Describe the difference between a vector and a ra. EXMLE for Es. 3 0 IMGE ND REIMGE Use the translation (, ) ( 2 8, 4). 3. What is the image of (2, 6)? 4. What is the image of (2, 5)? 5. What is the preimage of 9(23, 20)? 6. What is the preimage of D9(4, 23)? EXMLE 2 for Es. 4 GRHING N IMGE The vertices of n QR are (22, 3), Q(, 2), and R(3, 2). Graph the image of the triangle using prime notation. 7. (, ) ( 4, 6) 8. (, ) ( 9, 2 2) 9. (, ) ( 2 2, 2 5) 0. (, ) ( 2, 3) WRITING RULE n 999 is the image of n after a translation. Write a rule for the translation. Then verif that the translation is an isometr ERROR NLYSIS Describe and correct the error in graphing the translation of quadrilateral EGH. (, ) ( 2, 2 2) E H E H G G 4. MULTILE HOIE Translate Q(0, 28) using (, ) ( 2 3, 2). Q9(22, 5) Q9(3, 20) Q9(23, 26) D Q9(2, 2) EXMLE 3 for Es IDENTIYING VETORS Name the vector and write its component form R D T J 568 hapter 9 roperties of Transformations

8 VETORS Use the point (23, 6). ind the component form of the vector that describes the translation to (0, ) 9. 9(24, 8) 20. 9(22, 0) 2. 9(23, 25) TRNSLTIONS Think of each translation as a vector. Describe the vertical component of the vector. Eplain EXMLE 4 for Es TRNSLTING TRINGLE The vertices of n DE are D(2, 5), E(6, 3), and (4, 0). Translate n DE using the given vector. Graph n DE and its image , , , , 24 LGER ind the value of each variable in the translation r 8 0 s 3w t 8 a8 b c LGER Translation maps (, ) to ( n, m). Translation maps (, ) to ( s, t). a. Translate a point using Translation, then Translation. Write a rule for the final image of the point. b. Translate a point using Translation, then Translation. Write a rule for the final image of the point. c. ompare the rules ou wrote in parts (a) and (b). Does it matter which translation ou do first? Eplain. 3. MULTI-STE ROLEM The vertices of a rectangle are Q(2, 23), R(2, 4), S(5, 4), and T(5, 23). a. Translate QRST 3 units left and 2 units down. ind the areas of QRST and Q9R9S9T9. b. ompare the areas. Make a conjecture about the areas of a preimage and its image after a translation. (l), K hoto/orbis; (r), Steve llen/icturequest 32. HLLENGE The vertices of n are (2, 2), (4, 2), and (3, 4). a. Graph the image of n after the transformation (, ) (, ). Is the transformation an isometr? Eplain. re the areas of n and n 999 the same? b. Graph a new triangle, n DE, and its image after the transformation given in part (a). re the areas of n DE and n D9E99 the same? 9. Translate igures and Use Vectors 569

9 ROLEM SOLVING EXMLE 2 for Es HOME DESIGN Designers can use computers to make patterns in fabrics or floors. On the computer, a cop of the design in Rectangle is used to cover an entire floor. The translation (, ) ( 3, ) maps Rectangle to Rectangle. 33. Use coordinate notation to describe the translations that map Rectangle to Rectangles, D, E, and. (0, 4) (3, 4) (0, 0) D E 34. Write a rule to translate Rectangle back to Rectangle. (0, 4) (6, 4) EXMLE 5 for Es SNOWSHOEING You are snowshoeing in the mountains. The distances in the diagram are in miles. Write the component form of the vector. 35. rom the cabin to the ski lodge 36. rom the ski lodge to the hotel 37. rom the hotel back to our cabin N W S E abin (0, 0) Ski lodge Hotel (, 2) (4, 2) HNG GLIDING hang glider travels from point to point D. t point, the hang glider changes direction, as shown in the diagram. The distances in the diagram are in kilometers. N D (22, 5) (9, 4) (0, 0) (7, ) 38. Write the component form for # z and # z 39. Write the component form of the vector that describes the path from the hang glider s current position to its intended destination D What is the total distance the hang glider travels? 4. Suppose the hang glider went straight from to D. Write the component form of the vector that describes this path. What is this distance? 42. EXTENDED RESONSE Use the equation a. Graph the line and its image after the translation 25, 4. What is an equation of the image of the line? b. ompare the line and its image. What are the slopes? the -intercepts? the -intercepts? c. Write an equation of the image of after the translation 2, 26 without using a graph. Eplain our reasoning See WORKED-OUT SOLUTIONS in Student Resources 5 STNDRDIZED TEST RTIE

10 43. SIENE 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. a. Describe the translation. b. Each grid square is 2 millimeters on a side. How far does the amoeba travel? c. Suppose the amoeba moves from 3 to G7 in 24.5 seconds. What is its speed in millimeters per second? DEGH MULTI-STE ROLEM You can write the equation of a parabola in the form 5 ( 2 h) 2 k, where (h, k) is the verte of the parabola. In the graph, an equation of arabola is 5 ( 2 ) 2 3, with verte (, 3). arabola 2 is the image of arabola after a translation. a. Write a rule for the translation. b. Write an equation of arabola 2. c. Suppose ou translate arabola using the vector 24, 8. Write an equation of the image. d. n equation of arabola 3 is 5 ( 5) Write a rule for the translation of arabola to arabola 3. Eplain our reasoning. (, 3) arabola arabola 2 (7, ) 45. TEHNOLOGY The standard form of an eponential equation is 5 a, where a > 0 and a Þ. Use the equation 5 2. a. Use a graphing calculator to graph 5 2 and Describe the translation from 5 2 to b. Use a graphing calculator to graph 5 2 and Describe the translation from 5 2 to HLLENGE Use properties of congruent triangles to prove part of the Translation Theorem, that a translation preserves angle measure. See EXTR RTIE in Student Resources ONLINE QUIZ at m.hrw.com 57

11 9.2 Use roperties of Matrices efore You performed translations using vectors. Now You will perform translations using matri operations. Wh? So ou can calculate the total cost of art supplies, as in E. 36. Ke Vocabular matri element dimensions.9-2.n.vm.8(+) dd, subtract, and multipl matrices of appropriate dimensions. RED VOULRY n element of a matri ma also be called an entr. matri is a rectangular arrangement of numbers in rows and columns. (The plural of matri is matrices.) Each number in a matri is called an element. 5 row column 7G The element in the second row and third column is 2. The dimensions of a matri are the numbers of rows and columns. The matri above has three rows and four columns, so the dimensions of the matri are (read 3 b 4 ). You can represent a figure in the coordinate plane using a matri with two rows. The first row has the -coordinate(s) of the vertices. The second row has the corresponding -coordinate(s). Each column represents a verte, so the number of columns depends on the number of vertices of the figure. E XMLE Represent figures using matrices Write a matri to represent the point or polgon. a. oint b. Quadrilateral D 2 D VOID ERRORS The columns in a polgon matri follow the consecutive order of the vertices of the polgon. Solution a. oint matri for b. olgon matri for D 24 0G -coordinate -coordinate -coordinates D G coordinates GUIDED RTIE for Eample. Write a matri to represent n with vertices (3, 5), (6, 7) and (7, 3). 2. How man rows and columns are in a matri for a heagon? 572 hapter 9 roperties of Transformations David Young-Wolff/Gett Images

12 DDING ND SUTRTING To add or subtract matrices, ou add or subtract corresponding elements. The matrices must have the same dimensions. E XMLE 2 dd and subtract matrices a G G (24)G G b G G (27) (22) 2 2 3G G TRNSLTIONS You can use matri addition to represent a translation in the coordinate plane. The image matri for a translation is the sum of the translation matri and the matri that represents the preimage. E XMLE 3 Represent a translation using matrices VOID ERRORS In order to add two matrices, the must have the same dimensions, so the translation matri here must have three columns like the polgon matri. The matri 5 3 represents n. ind the image matri that 0 2G represents the translation of n unit left and 3 units up. Then graph n and its image. Solution The translation matri is 3 3 3G. dd this to the polgon matri for the preimage to find the image matri G G G Translation olgon Image matri matri matri GUIDED RTIE for Eamples 2 and 3 In Eercises 3 and 4, add or subtract. 3. f 23 7g f 2 25g G G 5. The matri G represents quadrilateral JKLM. Write the translation matri and the image matri that represents the translation of JKLM 4 units right and 2 units down. Then graph JKLM and its image. 9.2 Use roperties of Matrices 573

13 USE NOTTION Recall that the dimensions of a matri are alwas written as rows 3 columns. MULTILYING MTRIES The product of two matrices and is defined onl when the number of columns in is equal to the number of rows in. If is an m 3 n matri and is an n 3 p matri, then the product is an m 3 p matri. p 5 (m b n) p (n b p) 5 (m b p) equal dimensions of You will use matri multiplication in later lessons to represent transformations. E XMLE 4 Multipl matrices Multipl 0 4 5G G. Solution The matrices are both 2 3 2, so their product is defined. Use the following steps to find the elements of the product matri. STE Multipl the numbers in the first row of the first matri b the numbers in the first column of the second matri. ut the result in the first row, first column of the product matri G 2 23 (2) 0(2)? 5 2 8G?? G STE 2 Multipl the numbers in the first row of the first matri b the numbers in the second column of the second matri. ut the result in the first row, second column of the product matri G 2 23 (2) 0(2) (23) 0(8) 5 2 8G?? G STE 3 Multipl the numbers in the second row of the first matri b the numbers in the first column of the second matri. ut the result in the second row, first column of the product matri G 2 23 (2) 0(2) (23) 0(8) 5 2 8G 4(2) 5(2)? G STE 4 Multipl the numbers in the second row of the first matri b the numbers in the second column of the second matri. ut the result in the second row, second column of the product matri G 2 23 (2) 0(2) (23) 0(8) 5 2 8G 4(2) 5(2) 4(23) 5(8)G STE 5 Simplif the product matri. (2) 0(2) (23) 0(8) 5 4(2) 5(2) 4(23) 5(8)G 2 23 at m.hrw.com 3 28G 574 hapter 9 roperties of Transformations

14 E XMLE 5 Solve a real-world problem SOTLL Two softball teams submit equipment lists for the season. bat costs $20, a ball costs $5, and a uniform costs $40. Use matri multiplication to find the total cost of equipment for each team. NOTHER WY You could solve this problem arithmeticall, multipling the number of bats b the price of bats, and so on, then adding the costs for each team. Solution irst, write the equipment lists and the costs per item in matri form. You will use matri multiplication, so ou need to set up the matrices so that the number of columns of the equipment matri matches the number of rows of the cost per item matri. EQUIMENT p OST 5 TOTL OST Dollars ats alls Uniforms Dollars ats 20 Women Women p alls 5 5 Men G Uniforms 40G Men?G? You can find the total cost of equipment for each team b multipling the equipment matri b the cost per item matri. The equipment matri is and the cost per item matri is 3 3, so their product is a 2 3 matri (20) 42(5) 6(40) G20 40G 5(20) 45(5) 8(40) G 5 245G 0 c The total cost of equipment for the women s team is $0, and the total cost for the men s team is $245. GUIDED RTIE for Eamples 4 and 5 Use the matrices below. Is the product defined? Eplain G 5 f 2 g G Multipl G G 0. f 5 g G 5 2G G 2. WHT I? In Eample 5, find the total cost for each team if a bat costs $25, a ball costs $4, and a uniform costs $ Use roperties of Matrices 575

15 9.2 EXERISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 3, 9, and 3 5 STNDRDIZED TEST RTIE Es. 2, 7, 24, 25, and 35 SKILL RTIE. VOULRY op and complete: To find the sum of two matrices, add corresponding?. 2. WRITING How can ou determine whether two matrices can be added? How can ou determine whether two matrices can be multiplied? EXMLE for Es. 3 6 USING DIGRM Use the diagram to write a matri to represent the given polgon. 3. n E 2 4. n ED 5. Quadrilateral DE 6. entagon DE E D EXMLE 2 for Es. 7 2 EXMLE 3 for Es. 3 7 MTRIX OERTIONS dd or subtract. 7. f 3 5g f 9 2g G G G G 0. f g 2 f g G G TRNSLTIONS ind the image matri that represents the translation of the polgon. Then graph the polgon and its image G ; 4 units up 4. G H J ; 2 units left and 2 3 2G 3 units down G G 5. L M N G ; 4 units right and 6. Q R S 25 0 ; 3 units right and 2 units up 4 2G unit down 7. MULTILE HOIE The matri that represents quadrilateral D is Which matri represents the image of the quadrilateral after G. translating it 3 units right and 5 units up? G G D G G 576 hapter 9 roperties of Transformations

16 EXMLE 4 for Es MTRIX OERTIONS Multipl. 8. f 5 2g 4 3G 9. f.2 3g G G 22. f 4 8 2g 3 2.5G G G 5G 2 G G 24. MULTILE HOIE Which product is not defined? 7 3 2G 5G 6 f 3 20g 30G G 6 D 4 0G OEN-ENDED MTH Write two matrices that have a defined product. Then find the product. 27Gf 5 5g 26. ERROR NLYSIS Describe and correct the error in the computation (26) 22(2) 5 4 0G 3 26G 4(3) 0(26)G TRNSLTIONS Use the described translation and the graph of the image to find the matri that represents the preimage units right and 2 units down units left and 5 units up W9 3 X9 Y9 D9 9 V9 Z9 29. MTRIX EQUTION Use the description of a translation of a triangle to find the value of each variable. Eplain our reasoning. What are the coordinates of the vertices of the image triangle? 2 2 w 27 v 27G 9 a b m cg n 29 3G 30. HLLENGE point in space has three coordinates (,, z), as shown at the right. rom the origin, a point can be forward or back on the -ais, left or right on the -ais, and up or down on the z-ais. a. You translate a point three units forward, four units right, and five units up. Write a translation matri for the point. b. You translate a figure that has five vertices. Write a translation matri to move the figure five units back, ten units left, and si units down. 3 z O 4 (3, 4, 5) Use roperties of Matrices 577

17 ROLEM SOLVING EXMLE 5 for E OMUTERS Two computer labs submit equipment lists. mouse costs $0, a package of Ds costs $32, and a keboard costs $5. Use matri multiplication to find the total cost of equipment for each lab. Lab 25 Mice 0 Ds 8 Keboards Lab 2 5 Mice 20 Ds 2 Keboards 32. SWIMMING Two swim teams submit equipment lists. The women s team needs 30 caps and 26 goggles. The men s team needs 5 caps and 25 goggles. cap costs $0 and goggles cost $5. a. Use matri addition to find the total number of caps and the total number of goggles for each team. b. Use matri multiplication to find the total equipment cost for each team. c. ind the total cost for both teams. MTRIX ROERTIES In Eercises 33 35, use matrices,, and G G G 33. MULTI-STE ROLEM Use the matrices above to eplore the ommutative ropert of Multiplication. a. What does it mean that multiplication is commutative? b. ind and compare and. c. ased on part (b), make a conjecture about whether matri multiplication is commutative. 34. MULTI-STE ROLEM Use the matrices above to eplore the ssociative ropert of Multiplication. a. What does it mean that multiplication is associative? b. ind and compare () and (). c. ased on part (b), make a conjecture about whether matri multiplication is associative. 35. SHORT RESONSE ind and compare ( ) and. Make a conjecture about matrices and the Distributive ropert RT Two art classes are buing supplies. brush is $4 and a paint set is $0. Each class has onl $225 to spend. Use matri multiplication to find the maimum number of brushes lass can bu and the maimum number of paint sets lass can bu. Eplain. 5 See WORKED-OUT SOLUTIONS in Student Resources 5 STNDRDIZED TEST RTIE lass brushes 2 paint sets lass 8 brushes paint sets hotodisc/gett Images

18 37. HLLENGE The total United States production of corn was 8,967 million bushels in 2002, and 0,4 million bushels in The table shows the percents of the total grown b four states. a. Use matri multiplication to find the number of bushels (in millions) harvested in each state each ear. Iowa % % b. How man bushels (in millions) were harvested Illinois 6.4% 7.9% in these two ears in Iowa? Nebraska 0.5%.% c. The price for a bushel of corn in Nebraska was $2.32 in 2002, and $2.45 in Use matri multiplication to find the total value of corn harvested in Nebraska in these two ears. Minnesota.7% 9.6% QUIZ. In the diagram shown, name the vector and write its component form. M Use the translation (, ) ( 3, 2 2). 2. What is the image of (2, 5)? 3. What is the image of (6, 3)? 4. What is the preimage of (24, 2)? L dd, subtract, or multipl G G G G 5G G See EXTR RTIE in Student Resources ONLINE QUIZ at m.hrw.com 579

19 Investigating GeometrTIVITY Use before erform Reflections Reflections in the lane MTERILS graph paper straightedge ttend to precision. QUESTION What is the relationship between the line of reflection and the segment connecting a point and its image? E XLORE Graph a reflection of a triangle STE STE 2 STE H G Draw a triangle Graph (23, 2), (24, 5), and (22, 6). onnect the points to form n. Graph a reflection Reflect n in the -ais. Label points 9, 9, and 9 appropriatel. Draw segments Draw } 9, } 9, and } 9. Label the points where these segments intersect the -ais as, G, and H, respectivel. DRW ONLUSIONS Use our observations to complete these eercises. ind the lengths of } H and } H9, } G and } G9, and } and } 9. ompare the lengths of each pair of segments. 2. ind the measures of HG, G, and G. ompare the angle measures. 3. How is the -ais related to } 9, } 9, and } 9? 4. Use the graph at the right. a. } K 9L9 is the reflection of } K L in the -ais. op the diagram and draw } K 9L9. b. Draw } K K 9 and } LL9. Label the points where the segments intersect the -ais as J and M. c. How is the -ais related to } K K 9 and } LL9? 5. How is the line of reflection related to the segment connecting a point and its image? K 2 L 580 hapter 9 roperties of Transformations

20 9.3 erform Reflections efore You reflected a figure in the - or -ais. Now You will reflect a figure in an given line. Wh? So ou can identif reflections, as in Es Ke Vocabular line of reflection reflection.9-2.g.o.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometr software. Specif a sequence of transformations that will carr a given figure onto another. reflection is a transformation that uses a line like a mirror to reflect an image. The mirror line is called the line of reflection. reflection in a line m maps ever point in the plane to a point 9, so that for each point one of the following properties is true: If is not on m, then m is the perpendicular bisector of } 9, or If is on m, then 5 9. E XMLE 9 m oint not on m 9 m oint on m Graph reflections in horizontal and vertical lines The vertices of n are (, 3), (5, 2), and (2, ). Graph the reflection of n described. a. In the line n: 5 3 b. In the line m: 5 Solution a. oint is 2 units left of n, so its reflection 9 is 2 units right of n at (5, 3). lso, 9 is 2 units left of n at (, 2), and 9 is unit right of n at (4, ). b. oint is 2 units above m, so 9 is 2 units below m at (, 2). lso, 9 is unit below m at (5, 0). ecause point is on line m, ou know that 5 9. n m GUIDED RTIE for Eample Digital Vision/Gett Images Graph a reflection of n from Eample in the given line erform Reflections 58

21 E XMLE 2 Graph a reflection in 5 The endpoints of } G are (2, 2) and G(, 2). Reflect the segment in the line 5. Graph the segment and its image. Solution REVIEW SLOE The product of the slopes of perpendicular lines is 2. The slope of 5 is. The segment from to its image, } 9, is perpendicular to the line of reflection 5, so the slope of } 9 will be 2 (because (2) 5 2). rom, move.5 units right and.5 units down to 5. rom that point, move.5 units right and.5 units down to locate 9(2, 2). The slope of } GG9 will also be 2. rom G, move 0.5 units right and 0.5 units down to 5. Then move 0.5 units right and 0.5 units down to locate G9(2, ). 5 G G9 9 OORDINTE RULES You can use coordinate rules to find the images of points reflected in four special lines. KEY ONET or Your Notebook oordinate Rules for Reflections If (a, b) is reflected in the -ais, its image is the point (a, 2b). If (a, b) is reflected in the -ais, its image is the point (2a, b). If (a, b) is reflected in the line 5, its image is the point (b, a). If (a, b) is reflected in the line 5 2, its image is the point (2b, 2a). E XMLE 3 Graph a reflection in 5 2 Reflect } G from Eample 2 in the line 5 2. Graph } G and its image. Solution Use the coordinate rule for reflecting in 5 2. (a, b) (2b, 2a) 9 3 G (2, 2) 9(22, ) G(, 2) G9(22, 2) G at m.hrw.com GUIDED RTIE for Eamples 2 and 3 4. Graph n with vertices (, 3), (4, 4), and (3, ). Reflect n in the lines 5 2 and 5. Graph each image. 5. In Eample 3, verif that } 9 is perpendicular to hapter 9 roperties of Transformations

22 RELETION THEOREM Recall that the image of a translation is congruent to the original figure. The same is true for a reflection. THEOREM or Your Notebook THEOREM 9.2 Reflection Theorem reflection is an isometr. m n > n 999 WRITE ROOS Some theorems, such as the Reflection Theorem, have more than one case. To prove this tpe of theorem, each case must be proven. ROVING THE THEOREM To prove the Reflection Theorem, ou need to show that a reflection preserves the length of a segment. onsider a segment } Q that is reflected in a line m to produce } 9Q9. There are four cases to prove: m m m m ase and Q are on the same side of m. ase 2 and Q are on opposite sides of m. ase 3 lies on m, and } Q is not to m. ase 4 Q lies on m, and } Q m. E XMLE 4 ind a minimum distance RKING You are going to bu books. Your friend is going to bu Ds. Where should ou park to minimize the distance ou both will walk? Solution Reflect in line m to obtain 9. Then draw } 9. Label the intersection of } 9 and m as. ecause 9 is the shortest distance between and 9 and 5 9, park at point to minimize the combined distance,, ou both have to walk. 9 m GUIDED RTIE for Eample 4 6. Look back at Eample 4. nswer the question b using a reflection of point instead of point. 9.3 erform Reflections 583

23 RELETION MTRIX You can find the image of a polgon reflected in the -ais or -ais using matri multiplication. Write the reflection matri to the left of the polgon matri, then multipl. Notice that because matri multiplication is not commutative, the order of the matrices in our product is important. The reflection matri must be first followed b the polgon matri. KEY ONET or Your Notebook Reflection Matrices Reflection in the -ais 0 0 2G Reflection in the -ais G E XMLE 5 Use matri multiplication to reflect a polgon The vertices of n DE are D(, 2), E(3, 3), and (4, 0). ind the reflection of n DE in the -ais using matri multiplication. Graph n DE and its image. Solution STE Multipl the polgon matri b the matri for a reflection in the -ais G D E 3 4 Reflection matri 2 3 0G 5 olgon matri 2() 0(2) 2(3) 0(3) 2(4) 0(0) 0() (2) 0(3) (3) 0(4) (0)G 5 D9 E G Image matri STE 2 Graph n DE and n D9E99. E9 D9 D E 9 GUIDED RTIE for Eample 5 The vertices of n LMN are L(23, 3), M(, 2), and N(22, ). ind the described reflection using matri multiplication. 7. Reflect n LMN in the -ais. 8. Reflect n LMN in the -ais. 584 hapter 9 roperties of Transformations

24 9.3 EXERISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 5, 3, and 33 5 STNDRDIZED TEST RTIE Es. 2, 2, 25, and 40 SKILL RTIE. VOULRY What is a line of reflection? 2. WRITING Eplain how to find the distance from a point to its image if ou know the distance from the point to the line of reflection. RELETIONS Graph the reflection of the polgon in the given line. EXMLE for Es ais 4. -ais D 3 D ais D EXMLES 2 and 3 for Es D D 2. MULTILE HOIE What is the line of reflection for n and its image? 5 0 (the -ais) D EXMLE 5 for Es. 3 7 USING MTRIX MULTILITION Use matri multiplication to find the image. Graph the polgon and its image. 3. Reflect G in the -ais. 4. Reflect Q R S G in the -ais. 9.3 erform Reflections 585

25 INDING IMGE MTRIES Write a matri for the polgon. Then find the image matri that represents the polgon after a reflection in the given line. 5. -ais 6. -ais 7. -ais 2 D 8. ERROR NLYSIS Describe and correct the error in finding the image matri of n QR reflected in the -ais G G G MINIMUM DISTNE ind point on the -ais so is a minimum. 9. (, 4), (6, ) 20. (4, 23), (2, 25) 2. (28, 4), (2, 3) TWO RELETIONS The vertices of n GH are (3, 2), G(, 5), and H(2, 2). Reflect n GH in the first line. Then reflect n 9G9H9 in the second line. Graph n 9G9H9 and n 0G0H In 5 2, then in In 5 2, then in In 5, then in SHORT RESONSE Use our graphs from Eercises What do ou notice about the order of vertices in the preimages and images? 26. ONSTRUTION Use these steps to construct a reflection of n in line m using a straightedge and a compass. STE Draw n and line m. STE 2 STE 3 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 line m. Label that point 9. Repeat Step 2 to find points 9 and 9. Draw n 999. m 27. LGER The line is reflected in the line 5 2. What is the equation of the image? 28. LGER Reflect the graph of the quadratic equation in the -ais. What is the equation of the image? 29. RELETING TRINGLE Reflect n MNQ in the line HLLENGE oint 9(, 4) is the image of (3, 2) after a reflection in line c. Write an equation of line c N M See WORKED-OUT SOLUTIONS in Student Resources 5 STNDRDIZED TEST RTIE

26 ROLEM SOLVING RELETIONS Identif the case of the Reflection Theorem represented EXMLE 4 for E DELIVERING IZZ 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 total walking distance? 35. ROVING THEOREM 9.2 rove ase of the Reflection Theorem. ase The segment does not intersect the line of reflection. GIVEN c reflection in m maps to 9 and Q to Q9. ROVE c Q 5 9Q9 lan for roof a. Draw } 9, } QQ9, } RQ, and } RQ9. rove that n RSQ > n RSQ9. R 9 9 S m b. Use the properties of congruent triangles and perpendicular bisectors to prove that Q 5 9Q9. (r), Upperut Images/lam; (c), Gerald Hinde/Gett Images; (l), John Nakata/ Roalt-ree/orbis ROVING THEOREM 9.2 In Eercises 36 38, write a proof for the given case of the Reflection Theorem. (Refer to the diagrams in the theorem bo.) 36. ase 2 The segment intersects the line of reflection. GIVEN c reflection in m maps to 9 and Q to Q9. lso, } Q intersects m at point R. ROVE c Q 5 9Q9 37. ase 3 One endpoint is on the line of reflection, and the segment is not perpendicular to the line of reflection. GIVEN c reflection in m maps to 9 and Q to Q9. lso, lies on line m, and } Q is not perpendicular to m. ROVE c Q 5 9Q9 38. ase 4 One endpoint is on the line of reflection, and the segment is perpendicular to the line of reflection. GIVEN c reflection in m maps to 9 and Q to Q9. lso, Q lies on line m, and } Q is perpendicular to line m. ROVE c Q 5 9Q9 9.3 erform Reflections 587

27 39. RELETING OINTS Use (, 3). a. oint has coordinates (2, ). ind point on ] so 5. b. The endpoints of } G are (2, 0) and G(3, 2). ind point H on ] so 5 H. ind point J on G ] so G 5 J. c. Eplain wh parts (a) and (b) can be called reflection in a point. HYSIS The Law of Reflection states that the angle of incidence is congruent to the angle of reflection. Use this information in Eercises 40 and 4. angle of incidence angle of reflection 40. SHORT RESONSE Suppose a billiard table has a coordinate grid on it. If a ball starts at the point (0, ) and rolls at a 458 angle, it will eventuall return to its starting point. Would this happen if the ball started from other points on the -ais between (0, 0) and (0, 4)? Eplain. (0, 4) (8, 4) (0, ) (0, 0) (8, 0) 4. HLLENGE Use the diagram to prove that ou can see our full self in a mirror that is onl half of our height. ssume that ou and the mirror are both perpendicular to the floor. a. Think of a light ra starting at our foot and reflected in a mirror. Where does it have to hit the mirror in order to reflect to our ee? b. Think of a light ra starting at the top of our head and reflected in a mirror. Where does it have to hit the mirror in order to reflect to our ee? c. Show that the distance between the points ou found in parts (a) and (b) is half our height. D E 588 See EXTR RTIE in Student Resources ONLINE QUIZ at m.hrw.com

28 MIXED REVIEW of roblem Solving Make sense of problems and persevere in solving them.. MULTI-STE ROLEM n R9S9T9 is the image of n RST after a translation. T R S T9 R9 S9 a. Write a rule for the translation. b. Verif that the transformation is an isometr. c. Suppose n R9S9T9 is translated using the rule (, ) ( 4, 2 2). What are the coordinates of the vertices of n R99S99T99? 2. SHORT RESONSE During a marching band routine, a band member moves directl from point to point. Write the component form of the vector # z. Eplain our answer. 3. SHORT RESONSE Trace the picture below. Reflect the image in line m. How is the distance from X to line m related to the distance from X9 to line m? Write the propert that makes this true. X 5. GRIDDED NSWER The vertices of n GH are (24, 3), G(3, 2), and H(, 22). The coordinates of 9 are (2, 4) after a translation. What is the -coordinate of G9? 6. OEN-ENDED Draw a triangle in a coordinate plane. Reflect the triangle in an ais. Write the reflection matri that would ield the same result. 7. EXTENDED RESONSE Two cross-countr teams submit equipment lists for a season. pair of running shoes costs $60, a pair of shorts costs $8, and a shirt costs $5. Women s Team 4 pairs of shoes 6 pairs of shorts 6 shirts a. Use matri multiplication to find the total cost of equipment for each team. b. How much mone will the teams need to raise if the school gives each team $200? c. Repeat parts (a) and (b) if a pair of shoes costs $65 and a shirt costs $0. Does the change in prices change which team needs to raise more mone? Eplain. 8. MULTI-STE ROLEM Use the polgon as the preimage. E Men s Team 0 pairs of shoes 3 pairs of shorts 3 shirts D hillip Spears/Gett Images m 4. SHORT RESONSE The endpoints of } are (2, 4) and (4, 0). The endpoints of } D are (3, 3) and D(7, 2). Is the transformation from } to } D an isometr? Eplain. a. Reflect the preimage in the -ais. b. Reflect the preimage in the -ais. c. ompare the order of vertices in the preimage with the order in each image. Mied Review of roblem Solving 589

29 erform Rotations efore You rotated figures about the origin. Now You will rotate figures about a point. Wh? So ou can classif transformations, as in Es Ke Vocabular center of rotation angle of rotation rotation DIRETION O ROTTION Recall that a 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 8 maps ever point Q in the plane to a point Q9 so that one of the following properties is true: If Q is not the center of rotation, then Q 5 Q9 and m QQ9 5 8, or If Q is the center of rotation, then the image of Q is Q. 408 counterclockwise rotation is shown at the right. Rotations can be clockwise or counterclockwise. In this chapter, all rotations are counterclockwise. R R ngle of rotation enter of rotation clockwise E XMLE Draw a rotation Draw a 208 rotation of n about. counterclockwise.9-2.g.o.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometr software. Specif a sequence of transformations that will carr a given figure onto another. Solution STE Draw a segment from to. STE 2 Draw a ra to form a 208 angle with }. STE 3 Draw 9 so that STE 4 Repeat Steps 3 for each verte. Draw n hapter 9 roperties of Transformations harles E. efle/mira

30 USE ROTTIONS You can rotate a figure more than However, the effect is the same as rotating the figure b the angle minus ROTTIONS OUT THE ORIGIN You can rotate a figure more than 808. The diagram shows rotations of point 308, 2208, and 308 about the origin. rotation of 3608 returns a figure to its original coordinates. There are coordinate rules that can be used to find the coordinates of a point after rotations of 908, 808, or 2708 about the origin KEY ONET or Your Notebook 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 908, (a, b) (2b, a). 2. or a rotation of 808, (a, b) (2a, 2b). 3. or a rotation of 2708, (a, b) (b, 2a). (2b, a) (2a, 2b) (a, b) (b, 2a) E XMLE 2 Rotate a figure using the coordinate rules Graph quadrilateral RSTU with vertices R(3, ), S(5, ), T(5, 23), and U(2, 2). Then rotate the quadrilateral 2708 about the origin. NOTHER WY or an alternative method for solving the problem in Eample 2, see the roblem Solving Workshop. Solution Graph RSTU. Use the coordinate rule for a 2708 rotation to find the images of the vertices. (a, b) (b, 2a) R(3, ) R9(, 23) R S S(5, ) S9(, 25) T(5, 23) T9(23, 25) U9 U U(2, 2) U9(2, 22) R9 T Graph the image R9S9T9U9. T9 S9 at m.hrw.com GUIDED RTIE for Eamples and 2. Trace n DE and. Then draw a 508 rotation of n DE about. E 2. Graph n JKL with vertices J(3, 0), K(4, 3), and L(6, 0). Rotate the triangle 908 about the origin. D 9.4 erform Rotations 59

31 USING MTRIES You can find certain images of a polgon rotated about the origin using matri multiplication. Write the rotation matri to the left of the polgon matri, then multipl. KEY ONET or Your Notebook Rotation Matrices (ounterclockwise) 908 rotation 0 2 0G 808 rotation G RED VOULRY Notice that a 3608 rotation returns the figure to its original position. Multipling b the matri that represents this rotation gives ou the polgon matri ou started with, which is wh it is also called the identit matri rotation 0 2 0G 3608 rotation 0 0 G E XMLE 3 Use matrices to rotate a figure Trapezoid EGH has vertices E(23, 2), (23, 4), G(, 4), and H(2, 2). ind the image matri for a 808 rotation of EGH about the origin. Graph EGH and its image. VOID ERRORS ecause matri multiplication is not commutative, ou should alwas write the rotation matri first, then the polgon matri. Solution STE Write the polgon matri: E G H G STE 2 Multipl b the matri for a 808 rotation G E G H G E9 9 G9 H G Rotation olgon Image matri matri matri STE 3 Graph the preimage EGH. Graph the image E99G9H9. E H9 G9 G H E9 9 GUIDED RTIE for Eample 3 Use the quadrilateral EGH in Eample 3. ind the image matri after the rotation about the origin. Graph the image hapter 9 roperties of Transformations

32 THEOREM THEOREM 9.3 Rotation Theorem rotation is an isometr. or Your Notebook n > n 999 SES O THEOREM 9.3 To prove the Rotation Theorem, ou need to show that a rotation preserves the length of a segment. onsider a segment } QR rotated about point to produce } Q9R9. There are three cases to prove: R9 R9 R R R9 R ase R, Q, and are noncollinear. ase 2 R, Q, and are collinear. ase 3 and R are the same point. E XMLE 4 Standardized Test ractice The quadrilateral is rotated about. What is the value of? 8 } D Solution Theorem 9.3, the rotation is an isometr, so corresponding side lengths are equal. Then 2 5 6, so 5 3. Now set up an equation to solve for (3) Substitute 3 for. 5 2 Solve for. orresponding lengths in an isometr are equal. c The correct answer is. D GUIDED RTIE for Eample 4 6. ind the value of r in the rotation of the triangle D 5 2s 4r 2 3 3r s erform Rotations 593

33 9.4 EXERISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 3, 5, and 29 5 STNDRDIZED TEST RTIE Es. 2, 20, 2, 24, and 37 SKILL RTIE. VOULRY What is a center of rotation? 2. WRITING ompare the coordinate rules and the rotation matrices for a rotation of 908. EXMLE for Es. 3 IDENTIYING TRNSORMTIONS Identif the tpe of transformation, translation, reflection, or rotation, in the photo. Eplain our reasoning NGLE O ROTTION Match the diagram with the angle of rotation at m.hrw.com ROTTING IGURE Trace the polgon and point on paper. Then draw a rotation of the polgon the given number of degrees about G R EXMLE 2 for Es. 2 4 USING OORDINTE RULES Rotate the figure the given number of degrees about the origin. List the coordinates of the vertices of the image hapter 9 roperties of Transformations J J M K L R T S S 2 T (r), Digital Vision/Gett Images; (l), Roalt-ree/orbis; (c), Richard Naude/lam

34 EXMLE 3 for Es. 5 9 USING MTRIES ind the image matri that represents the rotation of the polgon about the origin. Then graph the polgon and its image G; J K L G; Q R S G; ERROR NLYSIS The endpoints of } are (2, ) and (2, 3). Describe and correct the error in setting up the matri multiplication for a 2708 rotation about the origin rotation of } 0 2 0G 2 2 3G 2708 rotation of } 2 2 3G 0 2 0G EXMLE 4 for Es MULTILE HOIE What is the value of in the rotation of the triangle about? } D MULTILE HOIE Suppose quadrilateral QRST is rotated 808 about the origin. In which quadrant is Q9? I II III D IV INDING TTERN The vertices of n are (2, 0), (3, 4), and (5, 2). Make a table to show the vertices of each image after a 908, 808, 2708, 3608, 4508, 5408, 6308, and 7208 rotation. What would be the coordinates of 9 after a rotation of 8908? Eplain. T S R 23. ONSTRUTION Use these steps to construct a rotation of n b angle D around a point O using a straightedge and a compass. STE Draw n, D, and O, the center of rotation. STE 2 Draw } O. Use the construction for coping an angle to cop D at O, as shown. Then use distance O and center O to find 9. STE 3 Repeat Step 2 to find points 9 and 9. Draw n 999. O D 24. SHORT RESONSE Rotate the triangle in Eercise about the origin. Show that corresponding sides of the preimage and image are perpendicular. Eplain. 25. VISUL RESONING point in space has three coordinates (,, z). What is the image of point (3, 2, 0) rotated 808 about the origin in the z-plane? (See Eercise 30 in the lesson Use roperties of Matrices.) HLLENGE Rotate the line the given number of degrees (a) about the -intercept and (b) about the -intercept. Write the equation of each image ; ; } 5; erform Rotations 595

35 ROLEM SOLVING NGLE O ROTTION Use the photo to find the angle of rotation that maps onto 9. Eplain our reasoning REVOLVING DOOR You enter a revolving door and rotate the door 808. What does this mean in the contet of the situation? Now, suppose ou enter a revolving door and rotate the door What does this mean in the contet of the situation? Eplain ROVING THEOREM 9.3 op and complete the proof of ase. ase The segment is noncollinear with the center of rotation. GIVEN c rotation about maps Q to Q9 and R to R9. ROVE c QR 5 Q9R9 STTEMENTS. Q 5 Q9, R 5 R9, m QQ9 5 m RR9 2. m QQ9 5 m QR9 m R9Q9 m RR9 5 m RQ m QR9 3. m QR9 m R9Q9 5 m RQ m QR9 4. m QR 5 m Q9R9 5.? >? 6. } QR > } Q9R9 7. QR 5 Q9R9 5 See WORKED-OUT SOLUTIONS in Student Resources RESONS. Definition of? 2.? 9 3.? ropert of Equalit 4.? ropert of Equalit 5. SS ongruence ostulate 6.? 7.? ROVING THEOREM 9.3 Write a proof for ase 2 and ase 3. (Refer to the diagrams below the theorem bo.) 34. ase 2 The segment is collinear with 35. ase 3 The center of rotation is one the center of rotation. endpoint of the segment. GIVEN c rotation about maps Q to GIVEN c rotation about maps Q Q9 and R to R9. to Q9 and R to R9., Q, and R are collinear. and R are the same point. ROVE c QR 5 Q9R9 5 STNDRDIZED TEST RTIE ROVE c QR 5 Q9R9 R9 R (tl) Dann Lehman/orbis; (tc) Tim Wright/orbis; (tr) Neil Rabinowitz/orbis; (cr) Image Source/Jupiterimages

36 36. MULTI-STE ROLEM Use the graph of a. Rotate the line 908, 808, 2708, and 3608 about the origin. Describe the relationship between the equation of the preimage and each image. 2 b. Do ou think that the relationships ou described in part (a) are true for an line? Eplain our reasoning. 37. EXTENDED RESONSE Use the graph of the quadratic equation 5 2 at the right. a. Rotate the parabola b replacing with and with in the original equation, then graph this new equation. 3 b. What is the angle of rotation? c. re the image and the preimage both functions? Eplain. TWO ROTTIONS The endpoints of } G are (, 2) and G(3, 4). Graph } 9G9 and } 99G99 after the given rotations. 38. Rotation: 908 about the origin 39. Rotation: 2708 about the origin Rotation: 808 about (0, 4) Rotation: 908 about (22, 0) 40. HLLENGE 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 (2, 308) at the right is 2 units from the origin and m XO What are the polar coordinates of the image of point after a 908 rotation? 808 rotation? 2708 rotation? Eplain X See EXTR RTIE in Student Resources ONLINE QUIZ at m.hrw.com 597

37 LESSON 9.4 Using LTERNTIVE METHODS nother Wa to Solve Eample 2 Make sense of problems and persevere in solving them. MULTILE RERESENTTIONS In Eample 2, ou saw how to use a coordinate rule to rotate a figure. You can also use tracing paper and move a cop of the figure around the coordinate plane. ROLEM Graph quadrilateral RSTU with vertices R(3, ), S(5, ), T(5, 23), and U(2, 2). Then rotate the quadrilateral 2708 about the origin. M ETHOD Using Tracing aper You can use tracing paper to rotate a figure. STE Graph the original figure in the coordinate plane. STE 2 Trace the quadrilateral and the aes on tracing paper. STE 3 Rotate the tracing paper Then transfer the resulting image onto the graph paper. R S R S U U T T RTIE. GRH Graph quadrilateral D with vertices (2, 22), (5, 23), (4, 25), and D(2, 24). Then rotate the quadrilateral 808 about the origin using tracing paper. 2. GRH Graph n RST with vertices R(0, 6), S(, 4), and T(22, 3). Then rotate the triangle 2708 about the origin using tracing paper. 3. SHORT RESONSE Eplain wh rotating a figure 908 clockwise is the same as rotating the figure 2708 counterclockwise. 4. SHORT RESONSE Eplain how ou could use tracing paper to do a reflection. 5. RESONING If ou rotate the point (3, 4) 908 about the origin, what happens to the -coordinate? What happens to the -coordinate? 6. GRH Graph n JKL with vertices J(4, 8), K(4, 6), and L(2, 6). Then rotate the triangle 908 about the point (2, 4) using tracing paper. 598 hapter 9 roperties of Transformations

38 Investigating GeometrTIVITY Use before ppl ompositions of Transformations Double Reflections MTERILS graphing calculator or computer onstruct viable arguments and critique the reasoning of others. QUESTION What happens when ou reflect a figure in two lines in a plane? E XLORE Double reflection in parallel lines STE Draw a scalene triangle onstruct a scalene triangle like the one at the right. Label the vertices D, E, and. STE 2 Draw parallel lines onstruct two parallel lines p and q on one side of the triangle. Make sure that the lines do not intersect the triangle. Save as EXLORE. STE 3 Reflect triangle Reflect n DE in line p. Reflect n D9E99 in line q. How is n D0E00 related to n DE? STE 4 Make conclusion Drag line q. Does the relationship appear to be true if p and q are not on the same side of the figure? EXLORE, STE 3 E XLORE 2 Double reflection in intersecting lines STE Draw intersecting lines ollow Step in Eplore for n. hange Step 2 from parallel lines to intersecting lines k and m. Make sure that the lines do not intersect the triangle. Label the point of intersection of lines k and m as. Save as EXLORE2. STE 2 Reflect triangle Reflect n in line k. Reflect n 999 in line m. How is n 000 related to n? STE 3 Measure angles Measure 0 and the acute angle formed b lines k and m. What is the relationship between these two angles? Does this relationship remain true when ou move lines k and m? EXLORE 2, STE 2 DRW ONLUSIONS Use our observations to complete these eercises. What other transformation maps a figure onto the same image as a reflection in two parallel lines? 2. What other transformation maps a figure onto the same image as a reflection in two intersecting lines? 3. Verif our conjectures in Eercises and 2 b using the appropriate transformation features of our graphing calculator or computer. 9.5 ppl ompositions of Transformations 599

39 9.5 ppl ompositions of Transformations efore You performed rotations, reflections, or translations. Now You will perform combinations of two or more transformations. Wh? So ou can describe the transformations that represent a rowing crew, as in E. 30. Ke Vocabular glide reflection composition of transformations translation followed b a reflection can be performed one after the other to produce a glide reflection. translation can be called a glide. glide reflection is a transformation in which ever point is mapped to a point 0 b the following steps STE irst, a translation maps to G.O.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometr software. Specif a sequence of transformations that will carr a given figure onto another. STE 2 Then, a reflection in a line k parallel to the direction of the translation maps 9 to 0. E XMLE ind the image of a glide reflection The vertices of n are (3, 2), (6, 3), and (7, ). ind the image of n after the glide reflection. k Translation: (, ) ( 2 2, ) Reflection: in the -ais Solution egin b graphing n. Then graph n 999 after a translation 2 units left. inall, graph n 000 after a reflection in the -ais. 9(26, 3) (6, 3) VOID ERRORS The line of reflection must be parallel to the direction of the translation to be a glide reflection. 9(29, 2) 0(29, 22) 9(25, ) 0(25, 2) 0(26, 23) (3, 2) (7, ) GUIDED RTIE for Eample. Suppose n in Eample is translated 4 units down, then reflected in the -ais. What are the coordinates of the vertices of the image? 600 hapter 9 roperties of Transformations 2. In Eample, describe a glide reflection from n 000 to n. Joel W. Rogers/orbis

40 OMOSITIONS When two or more transformations are combined to form a single transformation, the result is a composition of transformations. glide reflection is an eample of a composition of transformations. In this lesson, a composition of transformations uses isometries, so the final image is congruent to the preimage. This suggests the omposition Theorem. THEOREM or Your Notebook THEOREM 9.4 omposition Theorem The composition of two (or more) isometries is an isometr. E XMLE 2 ind the image of a composition The endpoints of } RS are R(, 23) and S(2, 26). Graph the image of } RS after the composition. Reflection: in the -ais Rotation: 908 about the origin VOID ERRORS Unless ou are told otherwise, do the transformations in the order given. Solution STE Graph } RS. STE 2 Reflect } RS in the -ais. } R9S9 has endpoints R9(2, 23) and S9(22, 26). STE 3 Rotate } R9S9 908 about the origin. } R0S0 has endpoints R0(3, 2) and S0(6, 22). 2 R9(2, 23) 25 S9(22, 26) R0(3, 2) R(, 23) S0(6, 22) S(2, 26) TWO RELETIONS ompositions of two reflections result in either a translation or a rotation, as described in the following two theorems. THEOREM or Your Notebook THEOREM 9.5 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 0 is the image of, then: k m 9. } 0 is perpendicular to k and m, and d, where d is the distance between k and m. 9 d ppl ompositions of Transformations 60

41 E XMLE 3 Use Theorem 9.5 In the diagram, a reflection in line k maps } GH to } G9H9. reflection in line m maps } G9H9 to } G0H0. lso, H 5 9 and DH a. Name an segments congruent to each segment: HG }, } H, and } G. b. Does 5 D? Eplain. c. What is the length of } GG0? Solution a. } HG > } H9G9, and } HG > } H0G0. } H > } H9. } G > } G9. b. Yes, 5 D because } GG0 and } HH0 are perpendicular to both k and m, so } D and } are opposite sides of a rectangle. c. the properties of reflections, H9 5 9 and H9D 5 4. Theorem 9.5 implies that GG0 5 HH0 5 2 p D, so the length of } GG0 is 2(9 4), or 26 units. H G k G9 H9 m D H0 G0 GUIDED RTIE for Eamples 2 and 3 3. Graph } RS from Eample 2. Do the rotation first, followed b the reflection. Does the order of the transformations matter? Eplain. 4. In Eample 3, part (c), eplain how ou know that GG0 5 HH0. Use the figure below for Eercises 5 and 6. The distance between line k and line m is.6 centimeters. 5. The preimage is reflected in line k, then in line m. Describe a single transformation that maps the blue figure to the green figure. 6. What is the distance between and 0? If ou draw 9 }, what is its relationship with line k? Eplain. k m 9 0 THEOREM or Your Notebook THEOREM 9.6 Reflections in Intersecting Lines Theorem If lines k and m intersect at point, then a reflection in k followed b a reflection in m is the same as a rotation about point. The angle of rotation is 28, where 8 is the measure of the acute or right angle formed b k and m. 0 0 m k m hapter 9 roperties of Transformations

42 E XMLE 4 Use Theorem 9.6 In the diagram, the figure is reflected in line k. The image is then reflected in line m. Describe a single transformation that maps to 0. Solution The measure of the acute angle formed between lines k and m is 708. So, b Theorem 9.6, a single transformation that maps to 0 is a 408 rotation about point. You can check that this is correct b tracing lines k and m and point, then rotating the point 408. at m.hrw.com m k GUIDED RTIE for Eample 4 7. In the diagram at the right, the preimage is reflected in line k, then in line m. Describe a single transformation that maps the blue figure onto the green figure. m 8. rotation of 768 maps to 9. To map to 9 using two reflections, what is the angle formed b the intersecting lines of reflection? 808 k 9.5 EXERISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 7, 7, and 27 5 STNDRDIZED TEST RTIE Es. 2, 25, 29, and 34 SKILL RTIE. VOULRY op and complete: In a glide reflection, the direction of the translation must be? to the line of reflection. EXMLE for Es WRITING Eplain wh a glide reflection is an isometr. GLIDE RELETION The endpoints of } D are (2, 25) and D(4, 0). Graph the image of } D after the glide reflection. 3. Translation: (, ) (, 2 ) 4. Translation: (, ) ( 2 3, ) Reflection: in the -ais Reflection: in Translation: (, ) (, 4) 6. Translation: (, ) ( 2, 2) Reflection: in 5 3 Reflection: in ppl ompositions of Transformations 603

43 EXMLE 2 for Es. 7 4 GRHING OMOSITIONS The vertices of n QR are (2, 4), Q(6, 0), and R(7, 2). Graph the image of n QR after a composition of the transformations in the order the are listed. 7. Translation: (, ) (, 2 5) 8. Translation: (, ) ( 2 3, 2) Reflection: in the -ais Rotation: 908 about the origin 9. Translation: (, ) ( 2, 4) 0. Reflection: in the -ais Translation: (, ) ( 2 5, 2 9) Rotation: 908 about the origin REVERSING ORDERS Graph } 0G 0 after a composition of the transformations in the order the are listed. Then perform the transformations in reverse order. Does the order affect the final image } 0G 0?. (25, 2), G(22, 4) 2. (2, 28), G(26, 23) Translation: (, ) ( 3, 2 8) Reflection: in the line 5 2 Reflection: in the -ais Rotation: 908 about the origin DESRIING OMOSITIONS Describe the composition of transformations D D9 D EXMLE 3 for Es. 5 9 USING THEOREM 9.5 In the diagram, k i m, n is reflected in line k, and n 999 is reflected in line m. 5. translation maps n onto which triangle? 6. Which lines are perpendicular to ] 0? 7. Name two segments parallel to ] 0. k 9 9 m If the distance between k and m is 2.6 inches, what is the length of } 0? 9. Is the distance from 9 to m the same as the distance from 0 to m? Eplain. 9 0 EXMLE 4 for Es USING THEOREM 9.6 ind the angle of rotation that maps onto m m 558 k 9 58 k See WORKED-OUT SOLUTIONS in Student Resources 5 STNDRDIZED TEST RTIE

44 22. ERROR NLYSIS student described the translation of } to 99 } followed b the reflection of 99 } to } 00 in the -ais as a glide reflection. Describe and correct the student s error USING MTRIES The vertices of n QR are (, 4), Q(3, 22), and R(7, ). Use matri operations to find the image matri that represents the composition of the given transformations. Then graph n QR and its image. 23. Translation: (, ) (, 5) 24. Reflection: in the -ais Reflection: in the -ais Translation: (, ) ( 2 9, 2 4) 25. OEN-ENDED MTH Sketch a polgon. ppl three transformations of our choice on the polgon. What can ou sa about the congruence of the preimage and final image after multiple transformations? Eplain. 26. HLLENGE The vertices of n JKL are J(, 23), K(2, 2), and L(3, 0). ind the image of the triangle after a 808 rotation about the point (22, 2), followed b a reflection in the line 5 2. ROLEM SOLVING EXMLE for Es NIML TRKS The left and right prints in the set of animal tracks can be related b a glide reflection. op the tracks and describe a translation and reflection that combine to create the glide reflection. 27. bald eagle (2 legs) 28. armadillo (4 legs) 8 in. 5 in. 29. MULTILE HOIE Which is not a glide reflection? The teeth of a closed zipper The tracks of a walking duck The kes on a computer keboard D The red squares on two adjacent rows of a checkerboard Greg ease/gett Images 30. ROWING Describe the transformations that are combined to represent an eight-person rowing shell. 9.5 ppl ompositions of Transformations 605

45 SWETER TTERNS In Eercises 3 33, describe the transformations that are combined to make each sweater pattern SHORT RESONSE Use the Reflections in arallel Lines Theorem to eplain how ou can make a glide reflection using three reflections. How are the lines of reflection related? 35. ROVING THEOREM 9.4 Write a plan for proof for one case of the omposition Theorem. GIVEN c rotation about maps Q to Q9 and R to R9. reflection in m maps Q9 to Q0 and R9 to R0. ROVE c QR 5 Q0R0 9 R9 R R0 m ROVING THEOREM 9.4 composition of a rotation and a reflection, as in Eercise 35, is one case of the omposition Theorem. List all possible cases, and prove the theorem for another pair of compositions. 37. ROVING THEOREM 9.5 rove the Reflection in arallel Lines Theorem. GIVEN c reflection in line l maps } JK to } J9K9, a reflection in line m maps } J9K9 to } J0K0, and l i m. ROVE c a. ] KK 0 is perpendicular to l and m. b. KK0 5 2d, where d is the distance between l and m. J K J9 K9 d m J 0 K ROVING THEOREM 9.6 rove the Reflection in Intersecting Lines Theorem. GIVEN c Lines k and m intersect at point. Q is an point not on k or m. ROVE c a. If ou reflect point Q in k, and then reflect its image Q9 in m, Q0 is the image of Q after a rotation about point. b. m QQ0 5 2(m ) lan for roof irst show k } QQ9 and } Q > } Q9. Then show n Q > n Q9. In the same wa, show n Q9 > n Q0. Use congruent triangles and substitution to show that } Q > } Q0. That proves part (a) b the definition of a rotation. Then use congruent triangles to prove part (b). 39. VISUL RESONING You are riding a biccle along a flat street. a. What two transformations does the wheel s motion use? b. Eplain wh this is not a composition of transformations. k 9 0 m STNDRDIZED TEST RTIE

46 40. MULTI-STE ROLEM point in space has three z coordinates (,, z). rom the origin, a point can be forward or back on the -ais, left or right on the -ais, and up or down on the z-ais. The endpoints O of segment } in space are (2, 0, 0) and (2, 3, 0), as shown at the right. a. Rotate } 908 about the -ais with center of rotation. What are the coordinates of } 99? b. Translate } 99 using the vector 4, 0, 2. What are the coordinates of } 00? 4. HLLENGE Justif the following conjecture or provide a countereample. onjecture When performing a composition of two transformations of the same tpe, order does not matter. QUIZ The vertices of n are (7, ), (3, 5), and (0, 7). Graph the reflection in the line.. -ais ind the coordinates of the image of (2, 23) after the rotation about the origin rotation rotation rotation The vertices of n QR are (28, 8), Q(25, 0), and R(2, 3). Graph the image of n QR after a composition of the transformations in the order the are listed. 7. Translation: (, ) ( 6, ) 8. Reflection: in the line 5 22 Reflection: in the -ais Rotation: 908 about the origin 9. Translation: (, ) ( 2 5, ) 0. Rotation: 808 about the origin Translation: (, ) ( 2, 7) Translation: (, ) ( 4, 2 3) See EXTR RTIE in Student Resources ONLINE QUIZ at m.hrw.com 607

47 Etension Tessellations GOL Make tessellations and discover their properties. Ke Vocabular tessellation tessellation is a collection of figures that cover a plane with no gaps or overlaps. You can use transformations to make tessellations..9-2.g.o.5 Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometr software. Specif a sequence of transformations that will carr a given figure onto another. regular tessellation is a tessellation of congruent regular polgons. In the figures above, the tessellation of equilateral triangles is a regular tessellation. E XMLE Determine whether shapes tessellate Does the shape tessellate? If so, tell whether the tessellation is regular. a. Regular octagon b. Trapezoid c. Regular heagon Solution VOID ERRORS The sum of the angles surrounding ever verte of a tessellation is This means that no regular polgon with more than si sides can be used in a regular tesssellation. a. regular octagon does not tessellate. b. The trapezoid tessellates. The tessellation is not regular because the trapezoid is not a regular polgon. gaps overlap c. regular heagon tessellates using translations. The tessellation is regular because it is made of congruent regular heagons. 608 hapter 9 roperties of Transformations

48 E XMLE 2 Draw a tessellation using one shape hange a triangle to make a tessellation. Solution STE STE 2 STE 3 ut a piece from the triangle. Slide the piece to another side. Translate and reflect the figure to make a tessellation. E XMLE 3 Draw a tessellation using two shapes Draw a tessellation using the given floor tiles. Solution RED VOULRY Notice that in the tessellation in Eample 3, the same combination of regular polgons meet at each verte. This tpe of tessellation is called semi-regular. STE STE 2 ombine one octagon and one square b connecting sides of the same length. Translate the pair of polgons to make a tessellation at m.hrw.com RTIE EXMLE for Es. 4 REGULR TESSELLTIONS Does the shape tessellate? If so, tell whether the tessellation is regular.. Equilateral triangle 2. ircle 3. Kite 4. OEN-ENDED MTH Draw a rectangle. Use the rectangle to make two different tessellations. Etension: Tessellations 609

49 5. MULTI-STE ROLEM hoose a tessellation and measure the angles at three vertices. a. What is the sum of the measures of the angles? What can ou conclude? b. Eplain how ou know that an quadrilateral will tessellate. EXMLE 2 for Es. 6 9 DRWING TESSELLTIONS In Eercises 6 8, use the steps in Eample 2 to make a figure that will tessellate. 6. Make a tessellation using a triangle as the base figure. 7. Make a tessellation using a square as the base figure. hange both pairs of opposite sides. 8. Make a tessellation using a heagon as the base figure. hange all three pairs of opposite sides. 9. ROTTION TESSELLTION Use these steps to make another tessellation based on a regular heagon DE. a. onnect points and with a curve. Rotate the curve 208 about so that coincides with. b. onnect points E and with a curve. Rotate the curve 208 about E so that coincides with D. c. onnect points and D with a curve. Rotate the curve 208 E about so that D coincides with. d. Use this figure to draw a tessellation. D EXMLE 3 for Es. 0 2 USING TWO OLYGONS Draw a tessellation using the given polgons OEN-ENDED MTH Describe three eamples of the use of tessellations in architecture, design, crafts, or other arts. TRNSORMTIONS Describe the transformation(s) used to make the tessellation USING SHES On graph paper, outline a capital H. Use this shape to make a tessellation. What transformations did ou use? 60 hapter 9 roperties of Transformations

50 9.6 Identif Smmetr efore You reflected or rotated figures. Now You will identif line and rotational smmetries of a figure. Wh? So ou can identif the smmetr in a bowl, as in E.. Ke Vocabular line smmetr line of smmetr rotational smmetr center of smmetr figure in the plane has line smmetr if 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 right. figure can have more than one line of smmetr. E XMLE Identif lines of smmetr How man lines of smmetr does the heagon have? m (bl), Rod lanck/hoto Researchers, Inc.; (bc), hotodisc/gett Images; (br), rco/.huetteer/lam; (t), Saulius T. Kondrotas/lam; (cr), James armichael Jr./NH/hotoshot.9-2.G.O.3 Given a rectangle, parallelogram, trapezoid, or regular polgon, describe the rotations and reflections that carr it onto itself. REVIEW RELETION Notice that the lines of smmetr are also lines of reflection. a. b. c. Solution a. Two lines of b. Si lines of c. One line of smmetr smmetr smmetr at m.hrw.com GUIDED RTIE for Eample How man lines of smmetr does the object appear to have? Draw a heagon with no lines of smmetr. 9.6 Identif Smmetr 6

51 REVIEW ROTTION or a figure with rotational smmetr, the angle of rotation is the smallest angle that maps the figure onto itself. ROTTIONL SYMMETRY figure in a plane has rotational smmetr if the figure can be mapped onto itself b a rotation of 808 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 908 or 808 maps the figure onto itself (although a rotation of 458 does not) The figure above also has point smmetr, which is 808 rotational smmetr. E XMLE 2 Identif rotational smmetr Does the figure have rotational smmetr? If so, describe an rotations that map the figure onto itself. a. arallelogram b. Regular octagon c. Trapezoid Solution a. The parallelogram has rotational smmetr. The center is the intersection of the diagonals. 808 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 458, 908, 358, or 808 about the center all map the octagon onto itself. c. The trapezoid does not have rotational smmetr because no rotation of 808 or less maps the trapezoid onto itself. GUIDED RTIE for Eample 2 Does the figure have rotational smmetr? If so, describe an rotations that map the figure onto itself. 5. Rhombus 6. Octagon 7. Right triangle 62 hapter 9 roperties of Transformations

52 E XMLE 3 Standardized Test ractice Identif the line smmetr and rotational smmetr of the equilateral triangle at the right. 3 lines of smmetr, 608 rotational smmetr 3 lines of smmetr, 208 rotational smmetr line of smmetr, 808 rotational smmetr D line of smmetr, no rotational smmetr ELIMINTE HOIES n equilateral triangle can be mapped onto itself b reflecting over an of three different lines. So, ou can eliminate choices and D. Solution The triangle has line smmetr. Three lines of smmetr can be drawn for the figure. or a figure with s lines of smmetr, the smallest rotation that maps the figure onto itself has the measure 3608 } s. So, the equilateral triangle has 3608 }, or 208 rotational smmetr. 3 c The correct answer is. D 208 GUIDED RTIE for Eample 3 8. Describe the lines of smmetr and rotational smmetr of a non-equilateral isosceles triangle. 9.6 EXERISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 7, 3, and 3 5 STNDRDIZED TEST RTIE Es. 2, 3, 4, 2, and 23 SKILL RTIE. VOULRY What is a center of smmetr? 2. WRITING Draw a figure that has one line of smmetr and does not have rotational smmetr. an a figure have two lines of smmetr and no rotational smmetr? EXMLE for Es. 3 5 LINE SYMMETRY How man lines of smmetr does the triangle have? Identif Smmetr 63

53 EXMLE 2 for Es. 6 9 ROTTIONL SYMMETRY Does the figure have rotational smmetr? If so, describe an rotations that map the figure onto itself EXMLE 3 for Es. 0 6 SYMMETRY Determine whether the figure has line smmetr and whether it has rotational smmetr. Identif all lines of smmetr and angles of rotation that map the figure onto itself MULTILE HOIE Identif the line smmetr and rotational smmetr of the figure at the right. line of smmetr, no rotational smmetr line of smmetr, 808 rotational smmetr No lines of smmetr, 908 rotational smmetr D No lines of smmetr, no rotational smmetr 4. MULTILE HOIE Which statement best describes the rotational smmetr of a square? The square has no rotational smmetr. The square has 908 rotational smmetr. The square has point smmetr. D oth and are correct. ERROR NLYSIS Describe and correct the error made in describing the smmetr of the figure The figure has line of smmetr and 808 rotational smmetr. DRWING IGURES In Eercises 7 20, use the description to draw a figure. If not possible, write not possible. 7. quadrilateral with no line 8. n octagon with eactl two lines of smmetr of smmetr 9. heagon with no point smmetr 20. trapezoid with rotational smmetr 5 See WORKED-OUT SOLUTIONS in Student Resources 5 STNDRDIZED TEST RTIE The figure has line of smmetr and 808 rotational smmetr. (c), Dann Lehman/orbis; (r), rtville; (l), Ja enni hotograph/hmh hoto

54 2. OEN-ENDED MTH Draw a polgon with 808 rotational smmetr and with eactl two lines of smmetr. 22. OINT SYMMETRY In the graph, } is reflected in the point to produce the image } 99. To make a reflection in a point for each point N on the preimage, locate N9 so that N9 5 N and N9 is on ] N. Eplain what kind of rotation would produce the same image. What kind of smmetr does quadrilateral 99 have? N 9 N SHORT RESONSE figure has more than one line of smmetr. an two of the lines of smmetr be parallel? Eplain. 24. RESONING How man lines of smmetr does a circle have? How man angles of rotational smmetr does a circle have? Eplain. 25. VISUL RESONING How man planes of smmetr does a cube have? 26. HLLENGE What can ou sa about the rotational smmetr of a regular polgon with n sides? Eplain. ROLEM SOLVING EXMLES and 2 for Es WORDS Identif the line smmetr and rotational smmetr (if an) of each word (bl), Larr Grant/Tai/Gett Images; (bc), David rausb/lam; (br), David Monette/lam KLEIDOSOES In Eercises 3 33, use the following information about kaleidoscopes. Inside a kaleidoscope, two mirrors are placed net to each other to form a V, as shown at the right. The angle between the mirrors determines the number of lines of smmetr in the image. Use the formula n(m ) to find the measure of between the mirrors or the number n of lines of smmetr in the image. alculate the angle at which the mirrors must be placed for the image of a kaleidoscope to make the design shown Identif Smmetr 65

55 34. HEMISTRY The diagram at the right shows two forms of the amino acid alanine. One form is laevo-alanine and the other is detro-alanine. How are the structures of these two molecules related? Eplain. 35. MULTI-STE ROLEM The astillo de San Marcos in St. ugustine, lorida, has the shape shown. z a. What kind(s) of smmetr does the shape of the building show? b. Imagine the building on a three-dimensional coordinate sstem. op and complete the following statement: The lines of smmetr in part (a) are now described as? of smmetr and the rotational smmetr about the center is now described as rotational smmetr about the?. 36. HLLENGE Spirals have a tpe of smmetr called spiral, or helical, smmetr. Describe the two transformations involved in a spiral staircase. Then eplain the difference in transformations between the two staircases at the right. 66 See EXTR RTIE in Student Resources ONLINE QUIZ at m.hrw.com (r), Jim Wark/Lonel lanet Images

56 Investigating Geometr ONSTRUTION Investigate Dilations MTERILS straightedge compass ruler Use before Identif and erform Dilations Use appropriate tools strategicall. QUESTION How do ou construct a dilation of a figure? dilation enlarges or reduces a figure to make a similar figure. You can use construction tools to make enlargement dilations. E XLORE onstruct an enlargement dilation Use a compass and straightedge to construct a dilation of n QR with a scale factor of 2, using a point outside the triangle as the center of dilation. STE STE 2 STE Q9 Q9 R R R9 R R9 Draw a triangle Draw n QR and choose the center of the dilation outside the triangle. Draw lines from through the vertices of the triangle. Use a compass Use a compass to locate 9 on ] so that 9 5 2(). Locate Q9 and R9 in the same wa. onnect points onnect points 9, Q9, and R9 to form n 9Q9R9. DRW ONLUSIONS Use our observations to complete these eercises. ind the ratios of corresponding side lengths of n QR and n 9Q9R9. re the triangles similar? Eplain. 2. Draw n DE. Use a compass and straightedge to construct a dilation with a scale factor of 3, using point D on the triangle as the center of dilation. 3. ind the ratios of corresponding side lengths of n DE and n D9E99. re the triangles similar? Eplain. 4. Draw n JKL. Use a compass and straightedge to construct a dilation with a scale factor of 2, using a point inside the triangle as the center of dilation. 5. ind the ratios of corresponding side lengths of n JKL and n J9K9L9. re the triangles similar? Eplain. 6. What can ou conclude about the corresponding angle measures of a triangle and an enlargement dilation of the triangle? 9.7 Identif and erform Dilations 67

57 9.7 Identif and erform Dilations efore You used a coordinate rule to draw a dilation. Now You will use drawing tools and matrices to draw dilations. Wh? So ou can determine the scale factor of a photo, as in E. 37. Ke Vocabular scalar multiplication dilation reduction enlargement.9-2.g.srt. Verif eperimentall the properties of dilations given b a center and a scale factor; a. dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given b the scale factor. dilation is a transformation in which the original figure and its image are similar. dilation with center and scale factor k maps ever point in a figure to a point 9 so that one of the following statements is true: If is not the center point, then the image point 9 lies on ]. The scale factor k is a positive number such that k 5 9 } and k Þ, or If is the center point, then R The dilation is a reduction if 0 < k < and it is an enlargement if k >. E XMLE Identif dilations ind the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. a. b R Solution a. ecause 9 } 5 } 2 8, the scale factor is k 5 } 3. The image 9 is 2 an enlargement. b. ecause 9 } 5 } 8 30, the scale factor is k 5 } 3. The image 9 is a reduction hapter 9 roperties of Transformations at m.hrw.com Raeanne Rubenstein/hotolibrar.com

58 E XMLE 2 Draw a dilation Draw and label ~ DEG. Then construct a dilation of ~ DEG with point D as the center of dilation and a scale factor of 2. Solution STE STE 2 STE 3 E E E9 D9 E D D D G G 9 G G9 G9 E9 9 Draw DEG. Draw ras from D through vertices E,, and G. Open the compass to the length of } DE. Locate E9 on DE ] so DE9 5 2(DE). Locate 9 and G9 the same wa. dd a second label D9 to point D. Draw the sides of D9E99G9. GUIDED RTIE for Eamples and 2. In a dilation, and 5 2. Tell whether the dilation is a reduction or an enlargement and find its scale factor. 2. Draw and label n RST. Then construct a dilation of n RST with R as the center of dilation and a scale factor of 3. MTRIES Scalar multiplication is the process of multipling each element of a matri b a real number or scalar. E XMLE 3 Scalar multiplication Simplif the product: G. Solution (3) 4(0) 4() G 4(2) 4(2) 4(23)G Multipl each element in the matri b G Simplif. GUIDED RTIE for Eample 3 Simplif the product G G 9.7 Identif and erform Dilations 69

59 DILTIONS USING MTRIES You can use scalar multiplication to represent a dilation centered at the origin in the coordinate plane. To find the image matri for a dilation centered at the origin, use the scale factor as the scalar. E XMLE 4 Use scalar multiplication in a dilation The vertices of quadrilateral KLMN are K(26, 6), L(23, 6), M(0, 3), and N(26, 0). Use scalar multiplication to find the image of KLMN after a dilation with its center at the origin and a scale factor of }. Graph KLMN 3 and its image. Solution K L } 3 K L M N Scale factor G K9 L9 M9 N olgon matri G Image matri N K9 N9 5 M L9 M9 E XMLE 5 ind the image of a composition The vertices of n are (24, ), (22, 2), and (22, ). ind the image of n after the given composition. Translation: (, ) ( 5, ) Dilation: centered at the origin with a scale factor of 2 Solution STE Graph the preimage n on the coordinate plane. 0(6, 6) STE 2 Translate n 5 units to the right and unit up. Label it n 999. STE 3 Dilate n 999 using the origin as the center and a scale factor of 2 to find n (2, 4) (6, 4) GUIDED RTIE for Eamples 4 and 5 5. The vertices of n RST are R(, 2), S(2, ), and T(2, 2). Use scalar multiplication to find the vertices of n R9S9T9 after a dilation with its center at the origin and a scale factor of segment has the endpoints (2, ) and D(, ). ind the image of } D after a 908 rotation about the origin followed b a dilation with its center at the origin and a scale factor of hapter 9 roperties of Transformations

60 9.7 EXERISES HOMEWORK KEY 5 See WORKED-OUT SOLUTIONS Es. 7, 9, and 35 5 STNDRDIZED TEST RTIE Es. 2, 24, 25, 27, 29, and 38 SKILL RTIE. VOULRY What is a scalar? EXMLE Es WRITING If ou know the scale factor, eplain how to determine if an image is larger or smaller than the preimage. IDENTIYING DILTIONS ind the scale factor. Tell whether the dilation is a reduction or an enlargement. ind the value of ERROR NLYSIS Describe and correct the error in finding the scale factor k of the dilation R9 R k 5 } 9 k 5 2 } EXMLE 2 for Es. 7 4 ONSTRUTION op the diagram. Then draw the given dilation. 7. enter H; k enter H; k enter J; k enter ; k 5 2 D E J. enter J; k 5 } 2 3. enter D; k 5 3 } 2 2. enter ; k 5 3 } 2 4. enter G; k 5 } 2 G H EXMLE 3 for Es. 5 7 EXMLE 4 for Es SLR MULTILITION Simplif the product G G G DILTIONS WITH MTRIES ind the image matri that represents a dilation of the polgon centered at the origin with the given scale factor. Then graph the polgon and its image. 8. D E k G; G H J G; k 5 } J L M N k 5 } G; Identif and erform Dilations 62

61 EXMLE 5 for Es OMOSING TRNSORMTIONS The vertices of n GH are (22, 22), G(22, 24), and H(24, 24). Graph the image of the triangle after a composition of the transformations in the order the are listed. 2. Translation: (, ) ( 3, ) Dilation: centered at the origin with a scale factor of Dilation: centered at the origin with a scale factor of } 2 Reflection: in the -ais 23. Rotation: 908 about the origin Dilation: centered at the origin with a scale factor of WRITING Is a composition of transformations that includes a dilation ever an isometr? Eplain. 25. MULTILE HOIE In the diagram, the center of the dilation of ~ QRS is point. The length of a side of ~ 9Q9R9S9 is what percent of the length of the corresponding side of ~ QRS? 9 S9 9 3 R9 S 2 9 R 25% 33% 300% D 400% 26. RESONING The distance from the center of dilation to the image of a point is shorter than the distance from the center of dilation to the preimage. Is the dilation a reduction or an enlargement? Eplain. 27. SHORT RESONSE Graph a triangle in the coordinate plane. Rotate the triangle, then dilate it. Then do the same dilation first, followed b the rotation. In this composition of transformations, does it matter in which order the triangle is dilated and rotated? Eplain our answer. 28. RESONING dilation maps (5, ) to 9(2, ) and (7, 4) to 9(6, 7). a. ind the scale factor of the dilation. b. ind the center of the dilation. 29. MULTILE HOIE Which transformation of (, ) is a dilation? (3, ) (2, 3) (3, 3) D ( 3, 3) 30. LGER Graph parabolas of the form 5 a 2 using three different values of a. Describe the effect of changing the value of a. Is this a dilation? Eplain. 3. RESONING Verif that the two figures in the graph are similar b describing a composition of transformations, involving a dilation then a translation, that maps nde to nd9e HLLENGE n has vertices (4, 2), (4, 6), and (7, 2). ind the vertices that represent a dilation of n centered at (4, 0) with a scale factor of 2. D9 D E E See WORKED-OUT SOLUTIONS in Student Resources 5 STNDRDIZED TEST RTIE

62 ROLEM SOLVING EXMLE for Es SIENE You are using magnifing glasses. Use the length of the insect and the magnification level to determine the length of the image seen through the magnifing glass. 33. Emperor moth 34. Ladbug 35. Dragonfl magnification 5 magnification 0 magnification mm 60 mm 47 mm 36. MURLS painter sketches plans for a mural. The plans are 2 feet b 4 feet. The actual mural will be 25 feet b 50 feet. What is the scale factor? Is this a dilation? Eplain. 37. HOTOGRHY adjusting the distance between the negative and the enlarged print in a photographic enlarger, ou can make prints of different sizes. In the diagram shown, ou want the enlarged print to be 9 inches wide (99). The negative is.5 inches wide (), and the distance between the light source and the negative is.75 inches (D). a. What is the scale factor of the enlargement? b. What is the distance between the negative and the enlarged print? 38. OEN-ENDED MTH Graph a polgon in a coordinate plane. Draw a figure that is similar but not congruent to the polgon. What is the scale factor of the dilation ou drew? What is the center of the dilation? (tl) Nature cutout s/lam; (tc) John T. owler/the Image inders; (tl) Rod lanck/hoto Researchers, Inc. 39. MULTI-STE ROLEM Use the figure at the right. a. Write a polgon matri for the figure. Multipl the matri b the scalar 22. b. Graph the polgon represented b the new matri. c. Repeat parts (a) and (b) using the scalar 2 } 2. d. Make a conjecture about the effect of multipling a polgon matri b a negative scale factor. 40. RE You have an 8 inch b 0 inch photo. a. What is the area of the photo? b. You photocop the photo at 50%. What are the dimensions of the image? What is the area of the image? c. How man images of this size would ou need to cover the original photo? H Identif and erform Dilations G

63 4. RESONING You put a reduction of a page on the original page. Eplain wh there is a point that is in the same place on both pages. 42. HLLENGE Draw two concentric circles with center. Draw } and } to the larger circle to form a 458 angle. Label points D and, where } and } intersect the smaller circle. Locate point E at the intersection of } and } D. hoose a point G and draw quadrilateral DEG. Use as the center of the dilation and a scale factor of }. Dilate DEG, n DE, and n E two times. 2 Sketch each image on the circles. Describe the result. G D E QUIZ Determine whether the figure has line smmetr and/or rotational smmetr. Identif the number of lines of smmetr and/or the rotations that map the figure onto itself Tell whether the dilation is a reduction or an enlargement and find its scale factor The vertices of n RST are R(3, ), S(0, 4), and T(22, 2). Use scalar multiplication to find the image of the triangle after a dilation centered at the origin with scale factor 4 } See EXTR RTIE in Student Resources ONLINE QUIZ at m.hrw.com

64 Technolog TIVITY ompositions with Dilations MTERILS graphing calculator or computer Use after Identif and erform Dilations m.hrw.com Kestrokes Use appropriate tools strategicall. QUESTION How can ou graph compositions with dilations? You can use geometr drawing software to perform compositions with dilations. E XMLE erform a reflection and dilation STE Draw triangle onstruct a scalene triangle like n at the right. Label the vertices,, and. onstruct a line that does not intersect the triangle. Label the line p. STE 2 Reflect triangle Select Reflection from the 4 menu. To reflect n in line p, choose the triangle, then the line. STES 2 STE 3 Dilate triangle Select Hide/Show from the 5 menu and show the aes. To set the scale factor, select lpha-num from the 5 menu, press ENTER when the cursor is where ou want the number, and then enter 0.5 for the scale factor. Net, select Dilation from the 4 menu. hoose the image of n, then choose the origin as the center of dilation, and finall choose 0.5 as the scale factor to dilate the triangle. Save this as DILTE. STE 3 RTIE. Move the line of reflection. How does the final image change? 2. To change the scale factor, select the lpha-num tool. lace the cursor over the scale factor. ress ENTER, then DELETE. Enter a new scale. How does the final image change? 3. Dilate with a center not at the origin. How does the final image change? 4. Use n and line p, and the dilation and reflection from the Eample. Dilate the triangle first, then reflect it. How does the final image change? 9.7 Identif and erform Dilations 625

65 MIXED REVIEW of roblem Solving Make sense of problems and persevere in solving them.. GRIDDED NSWER What is the angle of rotation, in degrees, that maps to 9 in the photo of the ceiling fan below? 9 5. MULTI-STE ROLEM The diagram shows the pieces of a puzzle SHORT RESONSE The vertices of n DE are D(23, 2), E(2, 3), and (3, 2). Graph n DE. Rotate n DE 908 about the origin. ompare the slopes of corresponding sides of the preimage and image. What do ou notice? 3. MULTI STE ROLEM Use pentagon QRST shown below. R T a. Write the polgon matri for QRST. b. ind the image matri for a 2708 rotation about the origin. c. Graph the image. 4. SHORT RESONSE Describe the transformations that can be found in the quilt pattern below. 626 hapter 9 roperties of Transformations S a. Which pieces are translated? b. Which pieces are reflected? c. Which pieces are glide reflected? 6. OEN-ENDED Draw a figure that has the given tpe(s) of smmetr. a. Line smmetr onl b. Rotational smmetr onl c. oth line smmetr and rotational smmetr 7. EXTENDED RESONSE In the graph below, n 999 is a dilation of n. 9 a. Is the dilation a reduction or an enlargement? b. What is the scale factor? Eplain our steps. c. What is the polgon matri? What is the image matri? d. When ou perform a composition of a dilation and a translation on a figure, does order matter? Justif our answer using the translation (, ) ( 3, 2 ) and the dilation of n. 9 9 (b), Scott erner/hotolibrar.com; (l), Thomas Holland

66 9 ig Idea HTER SUMMRY IG IDES or Your Notebook erforming ongruence and Similarit Transformations Translation Translate a figure right or left, up or down. Reflection Reflect a figure in a line m 9 Rotation Rotate a figure about a point. Dilation Dilate a figure to change the size but not the shape You can combine congruence and similarit transformations to make a composition of transformations, such as a glide reflection. ig Idea 2 Making Real-World onnections to Smmetr and Tessellations Line smmetr Rotational smmetr 4 lines of smmetr 908 rotational smmetr ig Idea 3 ppling Matrices and Vectors in Geometr You can use matrices to represent points and polgons in the coordinate plane. Then ou can use matri addition to represent translations, matri multiplication to represent reflections and rotations, and scalar multiplication to represent dilations. You can also use vectors to represent translations. hapter Summar 627

67 9 HTER REVIEW REVIEW KEY VOULRY m.hrw.com Multi-Language Glossar Vocabular practice or a list of postulates and theorems, see p. T2. image preimage isometr vector initial point, terminal point, horizontal component, vertical component component form matri element dimensions line of reflection center of rotation angle of rotation glide reflection composition of transformations line smmetr line of smmetr rotational smmetr center of smmetr scalar multiplication VOULRY EXERISES. op and complete: (n)? is a transformation that preserves lengths. 2. Draw a figure with eactl one line of smmetr. 3. WRITING Eplain how to identif the dimensions of a matri. Include an eample with our eplanation. Match the point with the appropriate name on the vector. 4. T. Initial point H 5. H. Terminal point T REVIEW EXMLES ND EXERISES Use the review eamples and eercises below to check our understanding of the concepts ou have learned in each lesson of this chapter. 9. Translate igures and Use Vectors E XMLE Name the vector and write its component form. The vector is E # z. rom initial point E to terminal point, ou move 4 units right and unit down. So, the component form is 4,. E EXMLES and 4 for Es. 6 7 EXERISES 6. The vertices of n are (2, 3), (, 0), and (22, 4). Graph the image of n after the translation (, ) ( 3, 2 2). 7. The vertices of n DE are D(26, 7), E(25, 5), and (28, 4). Graph the image of n DE after the translation using the vector 2, hapter 9 roperties of Transformations

68 m.hrw.com hapter Review ractice 9.2 Use roperties of Matrices E XMLE dd 5 24G 25G. These two matrices have the same dimensions, so ou can perform the addition. To add matrices, ou add corresponding elements G 5 25G G 6 2G EXMLE 3 for Es. 8 9 EXERISES ind the image matri that represents the translation of the polgon. Then graph the polgon and its image D E G G; G; 5 units up and 3 units left 2 units down 9.3 erform Reflections E XMLE The vertices of n MLN are M(4, 3), L(6, 3), and N(5, ). Graph the reflection of n MLN in the line p with equation 5 2. oint M is 2 units to the right of p, so its p reflection M9 is 2 units to the left of p at (0, 3). Similarl, L9 is 4 units to the left of p at (22, 3) L9 M9 M and N9 is 3 units to the left of p at (2, ). N9 5 2 N L EXMLES and 2 for Es. 0 2 EXERISES Graph the reflection of the polgon in the given line E J H 3 G L K hapter Review 629

69 9 HTER REVIEW 9.4 erform Rotations E XMLE ind the image matri that represents the 908 rotation of D about the origin The polgon matri for D is G. D Multipl b the matri for a 908 rotation G D G D G EXMLE 3 for Es. 3 4 EXERISES ind the image matri that represents the given rotation of the polgon about the origin. Then graph the polgon and its image. 3. Q R S L M N G; G; 9.5 ppl ompositions of Transformations E XMLE The vertices of n are (4, 24), (3, 22), and (8, 23). Graph the image of n after the glide reflection. Translation: (, ) (, 5) Reflection: in the -ais egin b graphing n. 0(23, 3) 9(3, 3) Then graph the image n 999 after a translation 0(28, 2) 9(8, 2) of 5 units up. inall, graph 0(24, ) 9(4, ) the image n 000 after a reflection in the -ais. (3, 22) (8, 23) (4, 24) EXMLE for Es. 5 6 EXERISES Graph the image of H(24, 5) after the glide reflection. 5. Translation: (, ) ( 6, 2 2) 6. Translation: (, ) ( 2 4, 2 5) Reflection: in 5 3 Reflection: in hapter 9 roperties of Transformations

70 m.hrw.com hapter Review ractice 9.6 Identif Smmetr E XMLE Determine whether the rhombus has line smmetr and/or rotational smmetr. Identif the number of lines of smmetr and/or the rotations that map the figure onto itself. The rhombus has two lines of smmetr. It also has rotational smmetr, because a 808 rotation maps the rhombus onto itself. EXMLES and 2 for Es. 7 9 EXERISES Determine whether the figure has line smmetr and/or rotational smmetr. Identif the number of lines of smmetr and/or the rotations that map the figure onto itself Identif and erform Dilations E XMLE Quadrilateral D has vertices (, ), (, 3), (3, 2), and D(3, ). Use scalar multiplication to find the image of D after a dilation with its center at the origin and a scale factor of 2. Graph D and its image. To find the image matri, multipl each element of the polgon matri b the scale factor. 9 2 D G D G Scale factor olgon matri Image matri 9 D 9 D9 EXMLE 4 for Es EXERISES ind the image matri that represents a dilation of the polgon centered at the origin with the given scale factor. Then graph the polgon and its image. Q R S L M N k 5 } 2 4 2G; G; k 5 3 hapter Review 63

71 9 HTER TEST Write a rule for the translation of n to n 999. Then verif that the translation is an isometr dd, subtract, or multipl G G G G G 3G Graph the image of the polgon after the reflection in the given line. 7. -ais D D ind the image matri that represents the rotation of the polgon. Then graph the polgon and its image. 0. n : rotation. KLMN: 2 5 G; 808 rotation G; The vertices of n QR are (25, ), Q(24, 6), and R(22, 3). Graph n 0Q0R0 after a composition of the transformations in the order the are listed. 2. Translation: (, ) ( 2 8, ) 3. Reflection: in the -ais Dilation: centered at the origin, k 5 2 Rotation: 908 about the origin Determine whether the flag has line smmetr and/or rotational smmetr. Identif all lines of smmetr and/or angles of rotation that map the figure onto itself hapter 9 roperties of Transformations

72 9 LGER REVIEW m.hrw.com MULTILY INOMILS ND USE QUDRTI ORMUL E XMLE Multipl binomials ind the product (2 3)( 2 7). Solution Use the OIL pattern: Multipl the irst, Outer, Inner, and Last terms. irst Outer Inner Last (2 3)( 2 7) 5 2() 2(27) 3() 3(27) Write the products of terms Multipl ombine like terms. E XMLE 2 Solve a quadratic equation using the quadratic formula Solve Solution Write the equation in standard form to be able to use the quadratic formula Write the original equation b 6 Ï} b 2 2 4ac }} 2a 2(25) 6 Ï}} (25) (2)() }}} 2(2) Ï} } 4 c The solutions are 5 Ï} 7 } Ï} 7 } 4 Write in standard form. Write the quadratic formula. Substitute values in the quadratic formula: a 5 2, b 5 25, and c 5. Simplif. ø 2.28 and 5 } 2 Ï} 7 ø EXMLE for Es. 9 EXMLE 2 for Es. 0 8 EXERISES ind the product.. ( 3)( 2 2) 2. ( 2 8) 2 3. ( 4)( 2 4) 4. ( 2 5)( 2 ) 5. (7 6) 2 6. (3 2 )( 9) 7. (2 )(2 2 ) 8. (23 ) 2 9. ( )(2 ) Use the quadratic formula to solve the equation lgebra Review 633

73 9 Standardized TEST RERTION Scoring Rubric ull redit solution is complete and correct artial redit solution is complete but has errors, or solution is without error but incomplete No redit no solution is given, or solution makes no sense SHORT RESONSE QUESTIONS ROLEM The vertices of n QR are (, 2), Q(4, 2), and R(0, 23). What are the coordinates of the image of n QR after the given composition? Describe our steps. Include a graph with our answer. Translation: (, ) ( 2 6, ) Reflection: in the -ais elow are sample solutions to the problem. Read each solution and the comments on the left to see wh the sample represents full credit, partial credit, or no credit. SMLE : ull credit solution The reasoning is correct, and the graphs are correct. irst, graph n QR. Net, to translate n QR 6 units left, subtract 6 from the -coordinate of each verte. R 0 2 (, 2) 9(25, 2) Q(4, 2) Q9(22, 2) R(0, 23) R9(26, 23) inall, reflect n 9Q9R9 in the -ais b multipling the -coordinates b 2. R9 R 9(25, 2) 0(25, ) Q9(22, 2) Q 0(22, ) R9(26, 23) R 0(26, 3) SMLE 2: artial credit solution Each transformation is performed correctl. However, the transformations are not performed in the order given in the problem. irst, graph n QR. Net, reflect n QR over the -ais b multipling each -coordinate b 2. inall, to translate n 9Q9R9 6 units left, subtract 6 from each -coordinate. The coordinates of the image of n QR after the composition are 0(25, ), Q 0(22, ), and R 0(26, 3). R 0 0 R9 2 0 R hapter 9 roperties of Transformations

74 SMLE 3: artial credit solution The reasoning is correct, but the student does not show a graph. irst subtract 6 from each -coordinate. So, 9( 2 6, 2) 5 9(25, 2), Q9(4 2 6, 2) 5 Q9(22, 2), and R9(0 2 6, 23) 5 R9(26, 23). Then reflect the triangle in the -ais b multipling each -coordinate b 2. So, 0(25, 2 p (2)) 5 0(25, ), Q 0(22, 2 p (2)) 5 Q 0(22, ), and R 0(26, 2 p (23)) 5 R 0(26, 3). SMLE 4: No credit solution The reasoning is incorrect, and the student does not show a graph. Translate n QR 6 units b adding 6 to each -coordinate. Then multipl each -coordinate b 2 to reflect the image over the -ais. The resulting n 9Q9R9 has vertices 9(27, 2), Q9(20, 2), and R9(26, 23). RTIE ppl Scoring Rubric Use the rubric on the previous page to score the solution to the problem below as full credit, partial credit, or no credit. Eplain our reasoning. ROLEM The vertices of D are (26, 2), (22, 3), (2, ), and D(25, ). Graph the reflection of D in line m with equation 5.. irst, graph D. ecause m is a vertical line, the reflection will not change the -coordinates. is 7 units left of m, so 9 is 7 units right of m, at 9(8, 2). Since is 3 units left of m, 9 is 3 units right of m, at 9(4, 3). The images of and D are 9(3, ) and D9(7, ). m 2 D 2 D 2. irst, graph D. The reflection is in a vertical line, so onl the -coordinates change. Multipl the -coordinates in D b 2 to get 9(6, 2), 9(2, 3), 9(, ), and D9(5, ). Graph 999D9. 2 D D Test reparation 635

75 9 Standardized TEST RTIE SHORT RESONSE. Use the square window shown below. 5. The design below is made of congruent isosceles trapezoids. ind the measures of the four interior angles of one of the trapezoids. Eplain our reasoning. a. Draw a sketch showing all the lines of smmetr in the window design. b. Does the design have rotational smmetr? If so, describe the rotations that map the design onto itself. 2. The vertices of a triangle are (0, 2), (2, 0), and (22, 0). What are the coordinates of the image of n after the given composition? Include a graph with our answer. 6. Two swimmers design a race course near a beach. The swimmers must move from point to point. Then the swim from point to point. inall, the swim from point to point D. Write the component form of the #z, and #z, vectors shown in the diagram, #z. Then write the component form of # D Dz. Dilation: (, ) (3, 3) Translation: (, ) ( 2 2, 2 2) 3. The light gra square is the image of the dark gra square after a single transformation. Describe three different transformations that could produce the image. D (24, 6) (9, 6) (7, 0) (0, 0) 7. polgon is reflected in the -ais and then reflected in the -ais. Eplain how ou can use a rotation to obtain the same result as this composition of transformations. Draw an eample. 8. In rectangle QRS, one side is twice as long costs $3.25, a soft drink costs $2.50, and a pretzel costs $3. The Johnson famil bus 5 hotdogs, 3 soft drinks, and pretzel. The Scott famil bus 4 hotdogs, 4 soft drinks, and 2 pretzels. Use matri multiplication to find the total amount spent b each famil. Which famil spends more mone? Eplain. 636 hapter 9 roperties of Transformations a. ind the lengths of the sides of QRS. Eplain. b. ind the ratio of the area of QRS to the area of 9Q9R9S9. Winchester Mster House 4. t a stadium concession stand, a hotdog as the other side. Rectangle 9Q9R9S9 is the image of QRS after a dilation centered at with a scale factor of 0.5. The area of 9Q9R9S9 is 32 square inches.

76 MULTILE HOIE 9. Which matri product is equivalent to the product f 3 2g 7 4G? f 23 g 4G 27 f 3g 7G 24 f 2 3g4G 7 D f 23g 27G 4 0. Which transformation is not an isometr? Translation Reflection Rotation D Dilation GRIDDED NSWER. Line p passes through points J(2, 5) and K(24, 3). Line q is the image of line p after line p is reflected in the -ais. ind the slope of line q. 2. The triangle on the left is the image of the triangle on the right after it is rotated about point. What is the value of? The vertices of n QR are (, 4), Q(2, 0), and R(4, 5). What is the -coordinate of Q9 after the given composition? Translation: (, ) ( 2 2, ) Dilation: centered at (0, 0) with k 5 2 EXTENDED RESONSE 4. n equation of line l is 5 3. a. Graph line l. Then graph the image of line l after it is reflected in the line 5. b. ind the equation of the image. c. Suppose a line has an equation of the form 5 a. Make a conjecture about the equation of the image of that line when it is reflected in the line 5. Use several eamples to support our conjecture. 5. The vertices of n EG are E(4, 2), (22, ), and G(0, 23). a. ind the coordinates of the vertices of n E99G9, the image of n EG after a dilation centered at the origin with a scale factor of 2. Graph n EG and n E99G9 in the same coordinate plane. b. ind the coordinates of the vertices of n E00G0, the image of n E99G9 after a dilation centered at the origin with a scale factor of 2.5. Graph n E00G0 in the same coordinate plane ou used in part (a). c. What is the dilation that maps n EG to n E00G0? d. What is the scale factor of a dilation that is equivalent to the composition of two dilations described below? Eplain. Dilation: centered at (0, 0) with a scale factor of a Dilation: centered at (0, 0) with a scale factor of b Test ractice 637