Performing Congruence and Similarity Transformations. C m

Transcription

1 9 ig Idea HPTER SUMMRY IG IES Performing ongruence and Similarit Transformations For Your Notebook 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. ilation ilate 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 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 635

2 9 REVIEW KEY VOULRY HPTER REVIEW classzone.com Multi-Language Glossar Vocabular practice For a list of postulates and theorems, see pp image, p. 57 preimage, p. 57 isometr, p. 573 vector, p. 574 initial point, terminal point, horizontal component, vertical component component form, p. 574 matri, p. 580 element, p. 580 dimensions, p. 580 line of reflection, p. 589 center of rotation, p. 598 angle of rotation, p. 598 glide reflection, p. 608 composition of transformations, p. 609 line smmetr, p. 69 line of smmetr, p. 69 rotational smmetr, p. 60 center of smmetr, p. 60 scalar multiplication, p. 67 VOULRY EXERISES. op and complete: (n)? is a transformation that preserves lengths.. raw 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 5. H. Terminal point H T REVIEW EXMPLES N EXERISES Use the review eamples and eercises below to check our understanding of the concepts ou have learned in each lesson of hapter Translate Figures and Use Vectors pp E X M P L E Name the vector and write its component form. The vector is# EF. z From initial point E to terminal point F, ou move 4 units right and unit down. So, the component form is 4,. E F EXMPLES and 4 on pp. 57, 574 for Es. 6 7 EXERISES 6. The vertices of n are (, 3), (, 0), and (, 4). Graph the image of n after the translation (, ) ( 3, ). 7. The vertices of nef are (6, 7), E(5, 5), and F(8, 4). Graph the image of nef after the translation using the vector, hapter 9 Properties of Transformations

3 classzone.com hapter Review Practice 9. Use Properties of Matrices pp E X M P L E ddf 5 4G F 5G. These two matrices have the same dimensions, so ou can perform the addition. To add matrices, ou add corresponding elements. F G F 5G 5F5 4 5G 5F 6 G EXMPLE 3 on p. 58 for Es. 8 9 EXERISES Find the image matri that represents the translation of the polgon. Then graph the polgon and its image. 8. F 8 9. F E F G G; G; 5 units up and 3 units left units down 9.3 Perform Reflections pp E X M P L E The vertices of nmln are M(4, 3), L(6, 3), and N(5, ). Graph the reflection of nmln in the line p with equation 5. Point M is units to the right of p, so its reflection M9 is units to the left of p at (0, 3). Similarl, L9 is 4 units to the left of p at (, 3) and N9 is 3 units to the left of p at (, ). L9 N9 M9 p M 5 N L EXMPLES and on pp for Es. 0 EXERISES Graph the reflection of the polgon in the given line E F J H 3 G L K hapter Review 637

4 9 Perform 9.4 HPTER REVIEW Rotations pp E X M P L E Find the image matri that represents the 908 rotation of about the origin. 3 The polgon matri for isf 4 4 G. Multipl b the matri for a 908 rotation. F 0 0GF G 5F G EXMPLE 3 on p. 600 for Es. 3 4 EXERISES Find the image matri that represents the given rotation of the polgon about the origin. Then graph the polgon and its image. 3. F Q R S F L M N P G; G; 9.5 ppl ompositions of Transformations pp E X M P L E The vertices of n are (4, 4), (3, ), and (8, 3). Graph the image of n after the glide reflection. Translation: (, ) (, 5) Reflection: in the -ais egin b graphing n. Then graph the image n 999 after a translation of 5 units up. Finall, graph the image n 000 after a reflection in the -ais. 0(8, ) 0(3, 3) 0(4, ) 9(3, 3) (3, ) 9(4, ) (4, 4) 9(8, ) (8, 3) EXMPLE on p. 608 for Es. 5 6 EXERISES Graph the image of H(4, 5) after the glide reflection. 5. Translation: (, ) ( 6, ) 6. Translation: (, ) ( 4, 5) Reflection: in 5 3 Reflection: in hapter 9 Properties of Transformations

5 classzone.com hapter Review Practice 9.6 Identif Smmetr pp E X M P L E etermine 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. EXMPLES and on pp for Es. 7 9 EXERISES etermine 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 Perform ilations pp E X M P L E Quadrilateral has vertices (0, 0), (0, 3), (, ), and (, 0). Use scalar multiplication to find the image of after a dilation with its center at the origin and a scale factor of. Graph and its image. To find the image matri, multipl each element of the polgon matri b the scale factor. Scale factor F G 5F Polgon matri 6 4 G Image matri EXMPLE 4 on p. 68 for Es. 0 EXERISES Find 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 0. F G; k 5 } 4. F 3 4G; k 5 3 hapter Review 639

6 9 Write HPTER TEST a rule for the translation of n to n 999. Then verif that the translation is an isometr dd, subtract, or multipl. 4. F G F G F 0 5 4GF 3G 0.8 4G F G F 7 3 Graph the image of the polgon after the reflection in the given line. 7. -ais Find the image matri that represents the rotation of the polgon. Then graph the polgon and its image. 0. n :F rotation. 5 G; KLMN:F 808 rotation 0 3 3G; The vertices of npqr are P(5, ), Q(4, 6), and R(, 3). Graph np0q0r0 after a composition of the transformations in the order the are listed.. Translation: (, ) ( 8, ) 3. Reflection: in the -ais ilation: centered at the origin, k 5 Rotation: 908 about the origin etermine 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 Properties of Transformations

7 9 LGER REVIEW MULTIPLY INOMILS N USE QURTI FORMUL lgebra classzone.com E X M P L E Multipl binomials Find the product ( 3)( 7). Solution Use the FOIL pattern: Multipl the First, Outer, Inner, and Last terms. First Outer Inner Last ( 3)( 7) 5 () (7) 3() 3(7) Write the products of terms Multipl. 5 ombine like terms. E X M P L E Solve a quadratic equation using the quadratic formula Solve 5 5. Solution Write the equation in standard form to be able to use the quadratic formula b 6 Ï} b 4ac }} a Write the original equation. Write in standard form. Write the quadratic formula. (5) 6 Ï}} (5) 5 4()() Substitute values in the quadratic formula: }}} () a 5, b 5 5, and c Ï} 5 8 } Ï} 7 } 4 Simplif. c The solutions are 5 Ï} 7 } 4 ø.8 and 5 Ï} 7 } 4 ø 0.. EXMPLE for Es. 9 EXMPLE for Es. 0 8 EXERISES Find the product.. ( 3)( ). ( 8) 3. ( 4)( 4) 4. ( 5)( ) 5. (7 6) 6. (3 )( 9) 7. ( )( ) 8. (3 ) 9. ( )( ) Use the quadratic formula to solve the equation lgebra Review 64

8 9 Standardized TEST PREPRTION Scoring Rubric Full redit solution is complete and correct Partial 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 RESPONSE QUESTIONS P RO L E M The vertices of npqr are P(, ), Q(4, ), and R(0, 3). What are the coordinates of the image of npqr after the given composition? escribe our steps. Include a graph with our answer. Translation: (, ) ( 6, ) Reflection: in the -ais elow are sample solutions to the problem. Read each solution and the comments in blue to see wh the sample represents full credit, partial credit, or no credit. SMPLE : Full credit solution The reasoning is correct, and the graphs are correct. First, graph npqr. Net, to translate npqr 6 units left, subtract 6 from the -coordinate of each verte. R 0 P(, ) P9(5, ) Q(4, ) Q9(, ) P 0 P9 Œ 0 Œ9 P Œ R(0, 3) R9(6, 3) Finall, reflect np9q9r9 in the -ais b multipling the -coordinates b. R9 R P9(5, ) P0(5, ) Q9(, ) Q0(, ) R9(6, 3) R0(6, 3) SMPLE : Partial credit solution Each transformation is performed correctl. However, the transformations are not performed in the order given in the problem. First, graph npqr. Net, reflect npqr over the -ais b multipling each -coordinate b. Finall, to translate np9q9r9 6 units left, subtract 6 from each -coordinate. The coordinates of the image of npqr after the composition are P0(, ), Q0(5, ), and R0(6, 3). R 0 P 0 R9 Œ 0 R P9 P Œ9 Œ 64 hapter 9 Properties of Transformations

9 SMPLE 3: Partial credit solution The reasoning is correct, but the student does not show a graph. First subtract 6 from each -coordinate. So, P9( 6, ) 5 P9(5, ), Q9(4 6, ) 5 Q9(, ), and R9(0 6, 3) 5 R9(6, 3). Then reflect the triangle in the -ais b multipling each -coordinate b. So, P0(5, p ()) 5 P0(5, ), Q0(, p ()) 5 Q0(, ), and R0(6, p (3)) 5 R0(6, 3). SMPLE 4: No credit solution The reasoning is incorrect, and the student does not show a graph. Translate npqr 6 units b adding 6 to each -coordinate. Then multipl each -coordinate b to reflect the image over the -ais. The resulting np9q9r9 has vertices P9(7, ), Q9(0, ), and R9(6, 3). PRTIE ppl Scoring Rubric Use the rubric on page 64 to score the solution to the problem below as full credit, partial credit, or no credit. Eplain our reasoning. PROLEM The vertices of are (6, ), (, 3), (, ), and (5, ). Graph the reflection of in line m with equation 5.. First, graph. 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, ). Since is 3 units left of m, 9 is 3 units right of m, at 9(4, 3). The images of and are 9(3, ) and 9(7, ). m. First, graph. The reflection is in a vertical line, so onl the -coordinates change. Multipl the -coordinates in b to get 9(6, ), 9(, 3), 9(, ), and 9(5, ). Graph Standardized Test Preparation 643

10 9 Standardized TEST PRTIE SHORT RESPONSE. Use the square window shown below. 5. The design below is made of congruent isosceles trapezoids. Find the measures of the four interior angles of one of the trapezoids. Eplain our reasoning. a. raw a sketch showing all the lines of smmetr in the window design. b. oes the design have rotational smmetr? If so, describe the rotations that map the design onto itself.. The vertices of a triangle are (0, ), (, 0), and (, 0). What are the coordinates of the image of n after the given composition? Include a graph with our answer. ilation: (, ) (3, 3) Translation: (, ) (, ) 3. The red square is the image of the blue square after a single transformation. escribe three different transformations that could produce the image. 4. t a stadium concession stand, a hotdog costs $3.5, a soft drink costs $.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 pretzels. Use matri multiplication to find the total amount spent b each famil. Which famil spends more mone? Eplain. 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. Finall, the swim from point to point. Write the component form of the vectors shown in the diagram,# z,# z, and. # z Then write the component form of# z. (0, 0) (9, 6) (7, 0) (4, 6) 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. raw an eample. 8. In rectangle PQRS, one side is twice as long as the other side. Rectangle P9Q9R9S9 is the image of PQRS after a dilation centered at P with a scale factor of 0.5. The area of P9Q9R9S9 is 3 square inches. a. Find the lengths of the sides of PQRS. Eplain. b. Find the ratio of the area of PQRS to the area of P9Q9R9S hapter 9 Properties of Transformations

11 STTE TEST PRTIE classzone.com MULTIPLE HOIE 9. Which matri product is equivalent to the product f 3 gf 7 4G? f3 gf 4G 7 f 3gF 7G 4 f 3gF4G 7 f 3gF 7G 4 0. Which transformation is not an isometr? Translation Rotation Reflection ilation GRIE NSWER. Line p passes through points J(, 5) and K(4, 3). Line q is the image of line p after line p is reflected in the -ais. Find the slope of line q.. The red triangle is the image of the blue triangle after it is rotated about point P. What is the value of? 5 P 4 3. The vertices of npqr are P(, 4), Q(, 0), and R(4, 5). What is the -coordinate of Q9 after the given composition? Translation: (, ) (, ) ilation: centered at (0, 0) with k 5 EXTENE RESPONSE 4. n equation of linel is 5 3. a. Graph linel. Then graph the image of linel after it is reflected in the line 5. b. Find 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 nefg are E(4, ), F(, ), and G(0, 3). a. Find the coordinates of the vertices of ne9f9g9, the image of nefg after a dilation centered at the origin with a scale factor of. Graph nefg and ne9f9g9 in the same coordinate plane. b. Find the coordinates of the vertices of ne0f0g0, the image of ne9f9g9 after a dilation centered at the origin with a scale factor of.5. Graph ne0f0g0 in the same coordinate plane ou used in part (a). c. What is the dilation that maps nefg to ne0f0g0? d. What is the scale factor of a dilation that is equivalent to the composition of two dilations described below? Eplain. ilation: centered at (0, 0) with a scale factor of a ilation: centered at (0, 0) with a scale factor of b Standardized Test Practice 645

12 UMULTIVE REVIEW 9 hapters 9 Tell whether the lines through the given points are parallel, perpendicular, or neither. (p. 7). Line : (3, 5), (, 6). Line : (, 0), (9, 8) Line : (3, 5), (4, 0) Line : (8, 6), (, 4) Write an equation of the line shown. (p. 80) 3. (4, 4) (4, ) 4. (, 4) 3 (0, ) 5. (, ) (0, ) State the third congruence that must be given to prove that the triangles are congruent using the given postulate or theorem. (pp. 34, 40, and 49) 6. SSS ongruence Post. 7. SS ongruence Post. 8. S ongruence Thm P R Y P S W V X Z etermine whether } is a perpendicular bisector, median, or altitude of n. (p. 39) etermine whether the segment lengths form a triangle. If so, would the triangle be acute, right, or obtuse? (pp. 38 and 44).,, , 44, , 9, , 8, , 40, ,.,.3 lassif the special quadrilateral. Eplain our reasoning. Then find the values of and. (p. 533) K 8 L J 5 M 0. X Y 5 8 (3 4)8 (5 5)8 (7 5)8 W Z 646 umulative Review: hapters 9

13 Graph the image of the triangle after the composition of the transformations in the order the are listed. (p. 608). P(5, ), Q(, 4), R(0, 0). F(, 8), G(6, 3), R(0, 0) Translation: (, ) (, 5) Reflection: in the line 5 Reflection: in the -ais Rotation: 908 about the origin FIRE ESPE In the diagram, the staircases on the fire escape are parallel. The measure of is 488. (p. 54) 3. Identif the angle(s) congruent to. 4. Identif the angle(s) congruent to. 5. What is m? 6. What is m 6? HM ISLNS The map of some of the ahamas has a scale of } inch : 60 miles. Use a ruler to estimate the actual distance from Freeport to Nassau. (p. 364) 8. NGLE OF ELEVTION You are standing feet awa from our house and the angle of elevation is 658 from our foot. How tall is our house? Round to the nearest foot. (p. 473) 9. PURSE You are decorating 8 trapezoid-shaped purses to sell at a craft show. You want to decorate the front of each purse with a string of beads across the midsegment. On each purse, the length of the bottom is 5.5 inches and the length of the top is 9 inches. If the beading costs $.59 per foot, how much will it cost to decorate the 8 purses? (p. 54) TILE PTTERNS escribe the transformations that are combined to make the tile pattern. (p. 607) umulative Review: hapters 9 647