Chapter 4. Geometry Resources by Chapter. Copyright Big Ideas Learning, LLC All rights reserved.

Transcription

1 Chapter Famil and Communit Involvement (English) Famil and Communit Involvement (Spanish) Section Section Section Section Section Section Cumulative Review

2 Name Date Chapter Transformations Dear Famil, Where do transformations occur in real life? Do the occur often? Your child ma have a difficult time recognizing transformations in objects he or she encounters ever da, but real-life transformations are more common than ou think. You and our child can work together to recognize everda transformations in our neighborhood. For instance, the traffic signs below show different tpes of transformations. SLOW XING R R Two-Wa Traffic Duck Crossing Railroad Crossing Roundabout What tpes of transformations are shown in the signs? Are the figures in the signs congruent? Are the similar? Take a walk in our neighborhood with our child and have him or her jot down or sketch an figures he or she thinks are possible transformations. Use the Internet or flip through a magazine to help identif compan logos that ma contain transformations. Some real-life transformations can be found quickl and easil, such as sliding open a window. Describe the tpes of transformations that our figures represent. Are the shapes that make up our figures congruent? Are the similar? How do ou know? Do our figures represent more than one transformation? Discuss whether a figure has an lines of smmetr or whether a logo has rotational smmetr. For eample, a logo at a gas station ma have rotational smmetr. Does our child like to pla video games? Have him or her create a logo for a fictional video game compan that uses one or more transformations. Be creative! 108

3 Nombre Fecha Capítulo Transformaciones Estimada familia: Dónde ocurren las transformaciones en la vida real? Se ocurren con frecuencia? A su hijo quizás le cueste reconocer las transformaciones en objetos que ve todos los días, pero las transformaciones en la vida real son más comunes que lo que creen. Usted su hijo pueden trabajar juntos para reconocer transformaciones cotidianas en su vecindario. Por ejemplo, las siguientes señales de tránsito muestran diferentes tipos de transformaciones. DESPACIO R R Tránsito de dos vías Cruce de patos Cruce de ferrocarril Rotonda Qué clases de transformaciones se muestran en las señales? Las figuras en las señales son congruentes? Son semejantes? Con su hijo, salgan a caminar por el vecindario pídale que anote o dibuje cualquier figura que crea que es una posible transformación. Usen Internet o una revista como auda para identificar logos de compañías que puedan contener transformaciones. Algunas transformaciones de la vida real pueden hallarse rápida fácilmente, tal como una traslación al abrir una ventana. Describan los tipos de transformaciones que representan sus figuras. Las formas que componen sus figuras son congruentes? Son semejantes? Cómo lo saben? Sus figuras representan más de una transformación? Comenten si una figura tiene algún eje de simetría o si un logo tiene simetría rotacional. Por ejemplo, un logo en una estación de gasolina podría tener simetría rotacional. A su hijo le gusta jugar videojuegos? Pídale que cree un logo para una compañía de videojuegos ficticia que tenga una o más transformaciones. Sean creativos! 109

4 .1 Start Thinking with coordinates ( ) ( ) ( ) Plot ABC A,, B,, C 1, 1, and A BC with coordinates A (, 0, ) B (,, ) C ( 1, 3) in a coordinate plane. Describe how to get from ABC to A BC. Compare the ordered pairs for each triangle visuall. Eplain how to use the ordered pairs for this eercise..1 Warm Up Translate point P. State the coordinates of P'. 1. P(, ); units down, units right. P( 3, ); 3 units right, 3 units up 3. P (, ); units down, units right. P( 1, ); 3 units left, 1 unit up 5. P(, 1 ); 1 unit up, units right 6. P( ) 6, 0 ; units up, units left.1 Cumulative Review Warm Up Prove the theorem. 1. Alternate Interior Angles Theorem (Theorem 3.). Alternate Eterior Angles Theorem (Theorem 3.3) 110

5 Name Date.1 Practice A 1. Name the vector and write its component form. K J. The vertices of ABC A, 3, B 1,, and C 0, 1. Translate ABC using the vector 1,. Graph ABC and its image. are ( ) ( ) ( ) 3. Find the component form of the vector that translates A( ) A ( ) 3, to 1,.. Write a rule for the translation of RST to R ST. S R S T R T 6 In Eercises 5 and 6, use the translation (, ) ( 1, 3) of the given point. 5. Q ( 5, 9) 6. M ( 3, 8) + to find the image with vertices ( ) ( ) ( ) In Eercises 7 and 8, graph CDE C 1, 3, D 0,, and E 1, 1 and its image after the given translation or composition. 7. Translation: (, ) ( 3, + 1) 8. Translation: (, ) ( + 10, 8) Translation: (, ) ( 7, + 15) 9. You want to plot the collinear points A(, 3 ), A (, ), and A ( 3, 7) on the same coordinate plane. Do ou have enough information to find the values of and? Eplain our reasoning. 10. You are using the map shown to navigate through the cit. You decide to walk to the Post Office from our current location at the Communit Center. Describe the translation that ou will follow. If each grid on the map is 0.05 mile, how far will ou travel? 3 Communit Center 1 Post Office

6 Name Date.1 Practice B 1. The vertices of FGH are F( ) G( ) H( ) using the vector, 7. Graph FGH and its image., 6, 3, 0, and 1,. Translate FGH. Find the component form of the vector that translates A( ) A ( ), 8 to 7, Write a rule for the translation of ABC to A BC. A B C A B C 8 In Eercises and 5, use the translation (, ) (, 3) image of the given point. + to find the. G(, ) 5. H ( 10, 5) 6. Graph JKL composition. with vertices J(, 8 ), K( 1, 3 ), and L( 5, ) Translation: (, ) ( + 6, 1) Translation: (, ) ( 1, 7) and its image after the 7. Is the transformation given b (, ) (, 1) our reasoning. + + a translation? Eplain 8. A popular kid s game has 15 tiles and 1 open space. The goal of the game is to rearrange the tiles to put them in order (from least to greatest, starting at the upper left-hand corner and going across each row). Use the figure to write the transformation(s) that describe the path of where the 8 tile is currentl, and where it must be b the end of the game. Can this same translation be used to describe the path of all the tiles? Graph an triangle and translate it in an direction. Draw translation vectors for each verte of the triangle. Is there a geometric relationship between all the translation vectors? Eplain wh this makes sense in terms of the slope of the line. 10. Point P(, ) undergoes a translation given b (, ) ( 3, a), another translation (, ) ( b, + 7) to produce the image of ( ) the values of a and b and point P. + followed b P 5, 8. Find 11

7 Name Date.1 Enrichment and Etension Properties of Vectors A two-dimensional vector V = a, b in standard position with its tail at ( 0, 0 ) has a horizontal component a and a vertical component b. The magnitude is the length of the line segment, given b V = a + b. Note: a and b ma also be denoted b V1 and V. 1. Find the magnitude of each vector. a. 5, 3 b. 3, 0 tail vector a head b c. head at ( 0, ) and tail at ( 3, ). Let U = a, b and V represents U + V. = c, d denote vectors in a plane. Write a vector that 3. Let U =, 3 and V = 6,. Write a vector that represents each of the following. a. U + V b. 3U c. V U d. U + V You are familiar with coordinates and vectors in the - coordinate plane, but in three dimensions, there are two other coordinate planes. There is the -z plane and the -z plane. Using the diagram, determine in which plane(s) ( -, z -, -z) each of the following points is located.. ( 3, 5, 0 ) 5. (, 0, 5 ) z 6. ( 0, 3, ) 7. ( 0, 3, 0 ) So, the magnitude of a vector V = V1, V, V3 in three dimensions is given b V = V1 + V + V3. 8. Let U = 1,, 0 and V = 5,, 3. Find the following. a. U b. V 113

8 Name Date.1 Puzzle Time What Can Go Up The Chimne Down, But Not Down The Chimne Up? Write the letter of each answer in the bo containing the eercise number. Complete the following questions. 1. What is another name for the original figure?. A translation is a? 3. A is a quantit that has both direction and magnitude, or size. Find the coordinates of the preimage.. (, ) ( 3, 5) A ( 3, 3 ) and B (, ) + with endpoints 5. (, ) ( 1, 3) A (, 0 ) and B ( 5, ) + with endpoints Find the rule for the translation of the coordinates. 6. A( 3, ) A ( 1, 8) B(, 6) B (, ) 7. A( 6, ) A (, ) B(, 5) B ( 6, 7) The vector 3, describes the translations ( ) ( ) ( ) ( ) A 1, A, 1 and B z 1, 1 B 3, Find the value of. 9. Find the value of. 10. Find the value of z Answers A. 7 I. (, ) ( +, ) Y. A( 3, 6 ), B(, 1) U. vector L. A( 6, ), B( 1, 1) N. (, ) ( +, + ) V. 1 E. A( 1, 3 ), B( 6, 7) F. 0 B. A( 0, 8 ), B( 5, 9) T. image M. (, ) (, ) G. L. 1 K. fleible motion R. preimage L. 1 O. line A. rigid motion P. (, ) (, ) 11

9 . Start Thinking La a ardstick at the base of a mirror. Stand at the end of the ardstick so ou are 3 feet from the mirror. Is our reflection the same distance from the mirror? Eplain wh or wh not. Hold up our right hand. Is our reflection holding up its right hand as well? Eplain wh or wh not.. Warm Up Reflect point P. State the coordinates of P'. 1. P( 5, 3 ); reflection in -ais. P(, 3 ); reflection in -ais 3. P( 1, 5 ); reflection in -ais. P( 1, 1 ); reflection in -ais 5. P (, 6 ); reflection in -ais 6. P ( 5, 1 ); reflection in -ais. Cumulative Review Warm Up Classif the angle

10 Name Date. Practice A In Eercises 1 3, graph ABC 1. ( ) ( ) ( ) A 0,, B 1, 3, C, ; -ais. ( ) ( ) ( ) A,, B 6,, C 3, 5 ; -ais A, 1, B 3, 8, C 1, 1 ; = 3. ( ) ( ) ( ) and its image after a reflection in the given line. In Eercises and 5, graph the polgon and its image after a reflection in the given line.. = 5. = P Q R 8 S P 8 S 8 R Q In Eercises 6 and 7, graph JKL its image after the glide reflection. with vertices J(, 3 ), K(, 1 ), and L( 1, 5) and 6. Translation: (, ) ( 1, ) 7. Translation: (, ) ( +, 3) Reflection: in the -ais Reflection: in the line = In Eercises 8 and 9, determine the number of lines of smmetr for the figure Find point W on the -ais so that VW XW X (, 1 ). + is a minimum given V (, 3) 11. A line = 3 5 is reflected in = a so that the image is given b = 1 3. What is the value of a? 1. Your friend claims that it is not possible to have a glide reflection if ou have two translations followed b one reflection. Is our friend correct? Eplain our reasoning. and 116

11 Name Date. Practice B In Eercises 1 and, graph CDE and its image after a reflection in the given line. 1. C( 3, ), D(, 1 ), E( 0, 5 ); -ais. ( ) ( ) ( ) C 1, 6, D 1,, E 7, 8; = 8 In Eercises 3 and, graph the polgon and its image after a reflection in the given line. 3. -ais. = 1 L 6 K 8 N K N M L M In Eercises 5 and 6, graph ABC its image after the glide reflection. with vertices A( 1, ), B(, 1 ), and C(, 3) and 5. Translation: (, ) ( +, 1) 6. Translation: (, ) ( 3, + 1) Reflection: in the line = Reflection: in the line = 7. Determine the number of lines of smmetr 8. Find point P on the -ais so that for the figure. AP + BP is a minimum. B A 9. Is it possible to perform two reflections of an object so that the final image is identical to the original image? If so, give an eample. If not, eplain our reasoning. 10. A triangle undergoes a glide reflection. Is it possible for the sides of the triangle to change length during this process? Eplain our reasoning. 11. Your friend claims that it is not possible to have a glide reflection if ou have one translation followed b two reflections. Is our friend correct? Eplain our reasoning. 117

12 Name Date. Enrichment and Etension Reflections 1. Reflect points F and G in the -ais. Name the coordinates and connect the points to form a polgon. Give the most specific name for the polgon.. Reflect points F and G in the -ais. Name the coordinates F(a, c) 8 G(a, b) 3. Reflect the points A and B in the line =. Connect the points to form a polgon. Give the most specific name for the polgon.. Reflect the points A and B in the line =. Connect the points to form a polgon. Give the most specific name for the polgon. A B are ( ) ( ) ( ) The vertices of ABC A,, B 0, 7, and C 1, 3. Reflect ABC in line 1 to obtain ABC. Then reflect ABC in line to obtain A B C. Graph triangles ABC and A B C. 5. Line 1: = ; Line : = 1 6. Line 1: = 3; Line : =

13 Name Date. Puzzle Time What Tpe Of Dance Does A Teacher Like? Circle the letter of each correct answer in the boes below. The circled letters will spell out the answer to the riddle. Complete the sentence. 1. A is a transformation that uses a line like a mirror to reflect the figure.. If ( a, b) is reflected in the -ais, then its image is the point. 3. If (, ) a b is reflected in the line =, then its image is the point.. A reflection is a transformation involving a translation followed b a reflection. 5. A figure in the plane has line when the figure can be mapped onto itself b a reflection in a line. How man lines of smmetr does the figure have? Identif the vertices of the image created after the reflection in the given line. 9. A( 3, ), B( 5, ); = 10. ( ) ( ) A 6, 3, B, ; -ais A, 1, B 3, 9 ; = 11. ( ) ( ) H S K L Q U W I A R E 9 ( b, a ) smmetr slider ( ) ( 3, 9) A 1,, B 5 ( 6, 3 ), (, ) A B G I D F O A D N E C E ( ) ( 5, ) A 3,, B ( b, a) ( a, b) 6.5 reflection rotation ( 1, ), ( 9, 3) A B 1 A B ( ) (, 5), 3, glide 119

14 .3 Start Thinking On a computer with a word processor, use 18-point Arial font to tpe the capital letters of the alphabet, putting a space between each letter. Which letter is smmetric when ou rotate the paper 90 degrees? Which letters are smmetric when ou rotate the paper 180 degrees? Are an letters not smmetric when ou rotate the paper 360 degrees?.3 Warm Up Rotate point P counterclockwise about the origin b the given angle. State the coordinates of P. 1. P (, ); 90. P ( 3, 0 ); P ( 6, 0 ); 180. P (, 6 ); P(, 0 ); P (, 0 ); 70.3 Cumulative Review Warm Up State the name of the propert. 1. For an segment AB, AB AB.. If A B, then B A. 10

15 Name Date.3 Practice A 1. Trace the polgon and point P. Then draw a 60 rotation of the polgon about point P. A P D B C. Graph the polgon and its image after a 70 rotation about the origin. J M K In Eercises 3 and, graph RST and its image after the composition. 3. Translation: (, ) (, 1) with vertices R(, 3 ), S(, 1 ), and T( 1, 5). Reflection: in the line = Rotation: 90 about the origin Rotation: 180 about the origin In Eercises 5 and 6, determine whether the figure has rotational smmetr. If so, describe an rotations that map the figure onto itself L 7. Draw AB with points A(, 0 ) and B ( 0, ). Rotate the segment 90 counterclockwise about point A. Then rotate the two segments 180 about the origin. What geometric figure did ou create using the original segment and its images? 8. List the uppercase letters of the alphabet that have rotational smmetr, and state the angle of the smmetr. 11

16 Name Date.3 Practice B 1. Graph the polgon and its image after a 90 rotation about the origin. W U T V 8 with vertices ( ) ( ) ( ) In Eercises and 3, graph CDE C 1, 3, D,, and E 5, 1 and its image after the composition.. Rotation: 180 about the origin 3. Reflection: in the line = Translation: (, ) ( 3, 1) + + Rotation: 70 about the origin In Eercises and 5, determine whether the figure has rotational smmetr. If so, describe an rotations that map the figure onto itself Is it possible to have an object that does not have 360 of rotational smmetr? Eplain our reasoning. 7. A figure that is rotated 60 is mapped back onto itself. Does the figure have rotational smmetr? Eplain. How man times can ou rotate the figure before it is back where it started? 8. Your friend claims that he can do a series of translations on an geometric object and get the same result as a rotation. Is our friend correct? 9. Your friend claims that she can do a series of reflections on an geometric object and get the same result as a rotation. Is our friend correct? 10. List the digits from 0 9 that have rotational smmetr, and state the angle of the smmetr. 1

17 Name Date.3 Enrichment and Etension Rotations In Eercises 1, rotate the line the given number of degrees about the given point. Write the equation of the image = 3; 90 ; -intercept. = + 8; 180 ; -intercept = 6; 90 ; -intercept. = + 5; 180 ; -intercept 5. In the diagram, A and B are the images of A and B after a 90 rotation about point P. a. Find the coordinates of A. b. Find the coordinates of B. c. The point ( 6, 1) is rotated 90 about (, 1 ). What are the coordinates of the image point? d. The point (, 5) is rotated 90 about ( 3, 7 ). What are the coordinates of the image of the point? B A B(a, b) P( 0, 0 ) A(a, 0 ) 6. The endpoints of FG are F( 1, ) and G( 3, ). Graph F G and F G after the given rotations. a. Rotation: 90 about the origin; Rotation: 180 about ( 0, ) b. Rotation: 70 about the origin; Rotation: 90 about (, 0) 13

18 Name Date.3 Puzzle Time What Did One Parallel Line Sa To The Other Parallel Line? A B C D E F G Complete each eercise. Find the answer in the answer column. Write the word under the answer in the bo containing the eercise letter. (1,1) MEET smmetric AND rotation WHAT (a, b) DOWN (3, 5) SKINNY (3, ) NEVER (b, a) WE Complete the sentence. A. A is a transformation in which a figure is turned about a fied point. B. When a point (a, b) is rotated counterclockwise about the origin for a rotation of 90, a, b. ( ) ( ) C. When a point (a, b) is rotated counterclockwise about the origin for a rotation of 180, a, b. ( ) ( ) D. When a point (a, b) is rotated counterclockwise about the origin for a rotation of 70, a, b. ( ) ( ) 1, 1. Find the verte of the image after a 70 rotation about the origin. Triangle ABC has vertices A( 3, 5 ), B(, 3 ), and C( ) E. A F. B G. C ( a, a) STRAIGHT ( a, b) SHAME (5, 3) WILL ( 1, 1) NAMED ( b, a) A (0, 0) DEEP (3, ) LONG 1

19 . Start Thinking Find at least two objects in each of the following categories: circle, square, triangle, and rectangle (nonsquare). Use a table to compare each object of the same categor in the following was: Are all angle measures the same? Is each shape eactl the same? Are the objects the same size?. Warm Up Plot and connect the points in a coordinate plane to make a polgon. Name the polgon. 1. A( 3, ), B(, 1 ), C( 3, 3). E( 1, ), F( 3, 1 ), G( 1, 3 ), H( 3, ) 3. J( 3, 3 ), K( 3, 3 ), L( 3, 3 ), M( 3, 3). P(,, ) Q(,, ) R( 5,, ) S(, ). Cumulative Review Warm Up Find the coordinates of point P along the directed line segment AB so that AP to PB is the given ratio. 1. A( ) B( ), 3, 9, 1 ; to 3. A( ) B( ) 1, 5, 7, 0; to 1 3. A( ) B( ) 1,,, 5 ; 3 to 1. A( ) B( ), 1, 6, 5 ; 3 to 5 15

20 Name Date. Practice A In Eercises 1 and, identif an congruent figures in the coordinate plane. Eplain In Eercises 3 and, describe a congruence transformation that maps ABC to ABC. 3.. A A 8 B A C B 8 B C C 8 B C 8 6 A In Eercises 5 and 6, determine whether the polgons with the given vertices are congruent. Use transformations to eplain our reasoning. 5. A( 5, ), B(, ), C(, 7 ) and S(, 5 ), T( 1, 5 ), U( 1, 0) 6. E( 6, ), F( 10, ), G( 10, 8 ), H( 6, 8 ) and W(, 8 ), X(, 10 ), Y( 8, 10 ), Z( 8, 8) 7. In the figure, a b, CDE is reflected in line a, and CDE is reflected in line b. List three pairs of segments that are parallel to each other. Then determine whether an segments are congruent to EE. In Eercises 8 and 9, find the measure of the acute or right angle formed b intersecting lines so that P can be mapped to P using two reflections. 8. A rotation of 8 maps P to P. 9. The rotation ( ) ( ),, maps P to P. D D C C C E E a b D E 16

21 Name Date. Practice B 1. Identif an congruent figures in the coordinate plane. Eplain Determine whether the polgons with the vertices A( 0, 6 ), B( 8, 6 ), C( 6, ), D (, ) and P( 3, ), Q( 7, ), R( 1, 8 ), S( 5, 8) are congruent. Use transformations to eplain our reasoning. In Eercises 3 5, JKL is reflected in line a, and JKL is reflected in line b. 3. JK is perpendicular to line a and has a length of 3 units, and verte K is 1 unit from line a. Find the distance JJ.. Find the angle of rotation that maps JKL onto J KL. 5. Is JK parallel to J K? Eplain our reasoning. 6. The rotation (, ) (, ) maps P and P. Find the measure of the acute or right angle formed b intersecting lines so that P can be mapped to P using two reflections. 7. Is it alwas, sometimes, or never true that the composition of two reflections results in the same image as a translation? Eplain our reasoning. J a K J K L L 80 L b K J 8. A is reflected in line s to form A and then reflected in line t to form A. Draw line t and intermediate A to complete the figure that represents these transformations. A s A 9. Your friend claims that if ou have a series of man parallel lines, reflecting a figure in two of the lines will produce the same result as reflecting the image in four or si of the lines. Is our friend correct? Eplain our reasoning. 17

22 Name Date. Enrichment and Etension Matri Addition and Translation A 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. The dimensions of a matri are the numbers of rows and columns. The matri to the right has three rows and four columns, so the dimensions of the matri are 3, read three b four You can represent a figure in the coordinate plane using a matri with two rows. The first row has the -coordinates of the vertices. The second row has the corresponding -coordinates. Each column represents a verte, so the number of columns depends on the number of vertices of the figure. Eample: Write a matri to represent point D. 1 Solution: The -coordinate is 1 and the -coordinate is 3. 3 D E 1. Write a matri to represent point F.. Write a polgon matri for DEFG. G 6 F To add or subtract matrices, ou add or subtract corresponding elements, and the resulting matri must have the same dimensions. Perform the operation In Eercises 5 8, use the diagram. 5. Write a polgon matri for ABC. 6. Write a matri that, when added to the polgon matri for ABC, translates the coordinates 1 unit left and 3 units up. This matri is called a translation matri. A C B 6 7. Add the translation matri from Eercise 6 to the polgon matri ABC. The result is called an image matri, which represents the sum of a translation matri and the matri of a preimage. 8. Graph a congruent triangle of ABC translated 1 unit left and 3 units up. Label the triangle A B C. What do ou notice about the resulting coordinates? 18

23 Name Date. Puzzle Time What Geometric Figure Is Like A Lost Parrot? Write the letter of each answer in the bo containing the eercise number. 1. Complete the sentence. Two geometric figures are figures if and onl if there is a rigid motion or a composition of rigid motions that maps one of the figures onto the other.. Congruent figures have the same size and shape. True or false? 3. Are three equilateral triangles with respective sides of 3 centimeters, centimeters, and inches congruent? Yes or no?. A figure is reflected in line k, and the image is then reflected in line m. The measure of the acute angle formed between lines k and m is. What is the angle of rotation? Given ABC with vertices A(, 3 ), B(, 3 ), and C(, 5 ), and the, +, 1, find the verte of the image. translation ( ) ( ) 5. A 6. B 7. C 8. Complete the sentence. The polgons with vertices A ( 0, 7 ), B( 0,, ) C( 5,, ) D ( 5, 7) and E( 7, 3 ), F( 7, 0 ), G ( 1, 0 ), H ( 1, 3 ) congruent. Answers R. are not I. constructed N. ( 6, 6) M. es A. are B. (, 6 ) G. (, ) L. 8 P. no E. 8 O. true X. (, ) Y. congruent O. ( 6, ) U. ( 6, 6)

24 .5 Start Thinking Shine a flashlight at a wall 6 feet awa in a diml lit room. Measure the diameter of the circle of light created on the wall. Move the flashlight 3 feet awa from the wall and measure the new diameter. Move the flashlight 1.5 feet awa from the wall and measure the diameter. Is there a pattern? Eplain what happens to the diameter of the circle of light as the flashlight is moved closer to the wall..5 Warm Up Use the graph to find the indicated length. 1. Find the length of BC.. Find the length of DE. A D E 6 G F B C.5 Cumulative Review Warm Up Plot the points in a coordinate plane. Then determine whether AB and CD are congruent. 1. A( 3, ), B( 3, 7 ), C( 3, ), D( 3, 1). A( 7,, ) B(,, ) C( 3,, ) D( 5, ) 3. A( 9,, ) B( 0,, ) C( 6, 9, ) D ( 6, ). A( 7, 9 ), B( 7, 0 ), C( 8, 3 ), D( 1, 3) 130

25 Name Date.5 Practice A In Eercises 1 and, find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement C P 7 P P 10 P C In Eercises 3 5, cop the diagram. Then use a compass and straightedge to construct a dilation of quadrilateral ABCD with the given center and scale factor k. 3. Center B, k = 3 A D P B C. Center P, k = 1 5. Center C, k = 75% In Eercises 6 and 7, graph the polgon and its image after a dilation with a scale factor k. P 1,, Q,, R,, S 1, 3 ; k = 6. ( ) ( ) ( ) ( ) A,, B, 6, C 1, 1, D, ; k = 75% 7. ( ) ( ) ( ) ( ) 8. A standard piece of paper is 8.5 inches b 11 inches. A piece of legal-size paper is 8.5 inches b 1 inches. B what scale factor k would ou need to dilate the standard paper so that ou could fit two pages on a single piece of legal paper? 9. The old film-stle cameras created photos that were best printed at 3.5 inches b 5 inches. Toda s new digital cameras create photos that are best printed at inches b 6 inches. Neither size picture will scale perfectl to fit in an 11-inch b 1-inch frame. Which tpe of camera will ou minimize the loss of the edges of our picture? 10. Your friend claims that if ou dilate a rectangle b a certain scale factor, then the area of the object also increases or decreases b the same amount. Is our friend correct? Eplain our reasoning. 1 in. 8.5 in. 11. Would it make sense to state A dilation has a scale factor of 1? Eplain our reasoning. 131

26 Name Date.5 Practice B In Eercises 1 and, find the scale factor of the dilation. Then tell whether the dilation is a reduction or an enlargement. 1.. P 9 P 1.5 C C 1 8 P P In Eercises 3 and, cop the diagram. Then use a compass and straightedge to construct a dilation with the given center and scale factor k. D A P B 3. Center B, k = C. Center P, k = 75% In Eercises 5 and 6, graph the polgon and its image after a dilation with a scale factor k. J 3,, K, 1, L 3,, M 5, ; k = 50% 5. ( ) ( ) ( ) ( ) V 1, 1, W 1, 0, X,, Y 3,, Z 0, 3; k = 3 6. ( ) ( ) ( ) ( ) ( ) 7. You look up at the sk at night and see the moon. It looks like it is about millimeters across. If ou then look at the moon through a telescope that has a magnification of 0 times, how big will it look to ou through the telescope? 8. What would it mean for an object to be dilated with a scale factor of k = 0? 9. Your friend claims that if ou dilate a rectangle b a certain scale factor, then the perimeter of the object also increases or decreases b the same factor. Is our friend correct? Eplain our reasoning. 10. The image shows an object that has been dilated with an unknown scale factor. Use the given measures to determine the scale factor and solve for the value of. A A C

27 Name Date.5 Enrichment and Etension Perimeter, Area, and Dilation Points A( 0, 0, ) B( 0,, ) C(, 0, ) and D(, ) form a rectangle on the - coordinate plane. 1. Plot points A, B, C, and D in a coordinate plane. Find the length, width, perimeter, and area of the rectangle, and then fill in the first row in the chart below. 1. Points A, B, C, and D. Points A, B, C, and D 3. Points D, E, F, and G Points A, B, C, and D are transformed under the operation (, ) (, ) generate points A, B, C, and D. Plot the new rectangle. Then find the new length, width, perimeter, and area, and fill in the second row in the chart above. are transformed under the operation ( ) ( ) Points A, B, C, and D,, to generate the points D, E, F, and G. Plot the new rectangle. Then find the new length, width, perimeter, and area, and fill in the last row in the chart above.. How does the transformation (, ) (, ) perimeter? area? affect the length and width? 3. A right triangle has vertices A( 0, 0 ), B( 10, 0 ), and C ( 10, ). How will the perimeter and area of the triangle change under the transformation,,? ( ) ( ). Write a general rule for the change in perimeter and area under the transformation, a, or, a, a. ( ) ( ) ( ) ( ) 5. Rectangle RSTU is defined b vertices R( 0, 0, ) S( 3, 0, ) T( 3, 5, ) and U( 0, 5. ) Write the transformation notation for RSTU area of 60 square units. Length Width Perimeter Area R S T U if the image has an 6. A microscope increases the side lengths of objects eight times. Calculate how big the area of a square will appear that has a side length of 0.6 millimeter. to 133

28 Name Date.5 Puzzle Time What Side Of A House Gets The Most Rain? Circle the letter of each correct answer in the boes below. The circled letters will spell out the answer to the riddle. Complete the sentence or solve the problem. 1. A is a transformation in which a figure is enlarged or reduced with respect to a fied point C, called the center, and a scale factor k, which is the ratio of the lengths of the corresponding sides of the image and the preimage.. When the scale factor k > 1, a dilation is a(n). 3. When 0 < k < 1, a dilation is a(n).. When a transformation changes the shape or size of a figure, the transformation is. 5. You want to reduce a picture that is 10 inches b 1 inches to a picture that is.5 inches b 3 inches. What is the scale factor k? 6. A magnifing glass shows the image of an object that is 10 times the object s actual size. Determine the length of the image of the object if the actual length of the object is 8 millimeters. 7. A magnifing glass shows the image of an object that is 6 times the object s actual size. Determine the actual length of the object if the image is 10 millimeters. Find the coordinates of the vertices after a dilation centered at the origin with scale factor k = A ( 3, 6) 9. B ( 3, 3) 10. C ( 9, 0) R T K L Q H E M A O ( 3, 6) 80 mm ( 9, 0) 0 epansion dilation ( 1, 1) alteration shrink reduction G I U T P S I N D E 8 ( 3, 0) 0 mm ( 1, 1 ) enlargement 1 rigid ( 1, ) nonrigid 13

29 .6 Start Thinking In a coordinate plane, draw an two squares. Label one ABCD and the other EFGH. Write down the coordinates for each verte. Using transformations and/or dilations, eplain how to find square EFGH beginning with square ABCD..6 Warm Up Solve. Round to the nearest tenth, if necessar n 1 w 3 =. = =. = c = 6. n = Cumulative Review Warm Up Decide whether inductive reasoning or deductive reasoning is used to reach the conclusion. Eplain. 1. Each time ou go to the store, ou spend mone. So, the net time ou go to the store, ou will spend mone.. Irrational numbers cannot be written as fractions. Rational numbers can be written as fractions. So, is a rational number. 3. All women are human. The first lad is a woman, so the first lad is human. 135

30 Name Date.6 Practice A In Eercises 1 and, graph PQR with vertices P( 1, 5 ), Q(, 3 ), and R(, 1) and its image after the similarit transformation. 1. Rotation: 180 about the origin. Dilation: (, ) ( 1, 1 ) Dilation: (, ) (, ) Reflection: in the -ais 3. Describe a similarit transformation that maps the black preimage onto the dashed image. X W W X 8 Y 8 Z Z Y In Eercises and 5, determine whether the polgons with the given vertices are similar. Use transformations to eplain our reasoning.. A(, 5 ), B(, ), C( 1, ) and 5. J( ) K( ) L( ) M( ) D( 3, 3 ), E( 3, 1 ), F(, 1) T( 3, 3, ) U(, 3, ) V(,, ) W( 3, 1) 5, 3, 3, 1, 3, 5, 5, 5 and 6. Prove that the figures are similar. Given Equilateral GHI with side length a, Prove equilateral PQR with side length b GHI is similar to PQR. I G a H P b Q R 7. Your friend claims ou can use a similarit transformation to turn a square into a rectangle. Is our friend correct? Eplain our answer. 8. Is the composition of a dilation and a translation commutative? In other words, do ou obtain the same image regardless of the order in which the transformations are performed? Justif our answer. 9. The image shown is known as a Sierpinski triangle. It is a common mathematical construct in the area of fractals. What can ou sa about the similarit transformations used to create the white triangles in this image? 136

31 Name Date.6 Practice B In Eercises 1 and, graph CDE with vertices C( 1, 3, ) D( 5, 3, ) and E(, 1) and its image after the similarit transformation. 1. Translation: (, ) ( 5, ). Reflection: in the -ais Dilation: (, ) ( 0.5, 0.5) Dilation: (, ) (, ) 3. Describe a similarit transformation that maps the black preimage onto the dashed image. 8 B C A D C D A B In Eercises and 5, determine whether the polgons with the given vertices are similar. Use transformations to eplain our reasoning.. A(, 0 ), B(, ), C(, 1 ) and 5. W( ) X( ) Y( ) Z( ) D(, 6 ), E(, ), F( 8, ) K( 0, 1 ), L( 5, ), M( 3, ), N( 1, ) 0, 1, 5, 1, 3,, 1, and 6. Prove that the figures are similar. Given: ABE DBC, AE CD A B C Prove: ABE is similar to DBC. E D 7. Is it possible to draw two circles that are not similar? Eplain our reasoning. 8. The image shows what tet often looks like when viewed through a magnifing glass. Does this represent a similarit transformation? Eplain our reasoning. 9. Your friend draws a sketch of triangles in his notebook like the one shown here. He then claims there are the same number of congruent triangles and similar triangles. Is our friend correct? Eplain. 137

32 Name Date.6 Enrichment and Etension Similarit Through the Origin If a figure is scaled b a factor of k about the origin, then the area of the new, similar image changes b a factor of k. Eample: A triangle has an area of 10 square units. A new triangle is mapped using, 5, 5. Find the area of the new triangle. ( ) ( ) Solution: If the new triangle is dilated b a factor of 5, then k = 5, and the new area will increase b a factor of 5 = 5. So, the new area will be 10 5 = 50 square units. 1. A dilated pentagon has an area of 60 square units after being mapped using,,. What was the original area? ( ) ( ). In the diagram, square ABCD has been enlarged through the origin b a factor of k. The resulting image is EFGH. What is the value of k? G 10 8 F 3. Calculate the area of ABCD. 6. Calculate the area of EFGH. H C B E 5. B what factor has the area of EFGH increased compared with ABCD? D A 6. In the diagram, Circle A is an enlargement of Circle B b a factor of k. The ratio of the area of Circle A to the area of Circle B is 9. The equations of the circles are as follows. Circle A: ( 1) ( ) t, where ( 1, ) + + = is the center of Circle A and t is the length of the radius. 6 B A Circle B: ( a) ( b) r, where ( a, b) + = is the center of Circle B and r is the length of the radius. a. What are the values of k, a, and b? b. What is the relationship between r and t? 138

33 Name Date.6 Puzzle Time Wh Did The Students Do Multiplication Problems On The Floor? A B C D E F G H Complete each eercise. Find the answer in the answer column. Write the word under the answer in the bo containing the eercise letter. false NOT not similar AND not maintain BECAUSE na TO alwas TABLES transitional FLOOR true CUSTODIAN similarit TOLD Complete the sentence. A. Two figures are figures when the have the same shape but not necessaril the same size. B. transformations preserve length and angle measure. C. transformations preserve angle measure onl. Determine whether the following are congruent. D. A( ) B( ) C( ) D( ) R( 0, 3 ), S(, 0 ), T(, 3 ), U(, 0) 5, 6, 3, 3, 7, 0, 9, 3 and Yes or no? E. A(,, ) B( 7,, ) C( 5,, ) D( 1, ) and R( 8, 8 ), S( 1, ), T( 10, ), U(, ) True or false? F. A( 3, 6 ), B( 6, 3 ), C( 3, 3 ) and R( 1, ), S(, 1 ), T( 1, 1) Yea or na? Answer the question. G. If a triangle is transformed b a dilation with a scale factor of 1, will it maintain congruenc or not maintain congruenc? H. Do similarit transformations preserve angle measure alwas or not alwas? dilation STAY congruence TEACHER not alwas CLASS es THEM ea GOT maintain USE no BAD similar THE 139

34 Name Date Chapter Cumulative Review In Eercises 1 1, simplif the epression ( + 11) 5. ( ) 6. 6( 6) 7. 3( + ) ( 1) ( 3) 10. ( + 15) + 8( 3) 11. 5( 1) 9( ) 1. 10( 5 + ) + 6( ) In Eercises 13 8, solve the equation. Check our solution = = = = = = = = = 11. = =. ( ) = ( + ) = ( ) = ( 7 + 6) = ( ) = You bu two tpes of fish at the local market. You need 1.5 pounds of tilapia and 1 pound of cod. Tilapia costs $3.88 per pound and cod costs $3.53 per pound. a. How much is our fish purchase? b. You give the cashier $0. How much change do ou receive? 30. You are making juice from concentrate. The directions on the packaging sa to mi 1 can of juice with 3 cans of water. A can is 1 fluid ounces. a. How man fluid ounces is the prepared juice? b. How man cups is the prepared juice (remember that 8 fluid ounces = 1 cup)? c. How man pints is the prepared juice (remember that cups = 1 pint)? d. How man quarts is the prepared juice (remember that cups = 1 quart)? 10

35 Name Date Chapter Cumulative Review (continued) In Eercises 31 3, tell whether the two figures are similar You want to decorate around the top of a jar with ribbon. The length around the jar is 18 inches. a. How man feet of ribbon do ou need? b. The ribbon costs $.80 per ard. How much does it cost per foot? c. According to how much ribbon ou need, how much will it cost? 36. You and our brother plan to fi a broken window in the garage door. The window measures 1 foot b 1.5 feet. a. What are the window measurements in inches? b. What is the area of the window in square inches? c. The price of glass is $0.03 per square inch. How much will the glass cost for the window? d. You pa with a $0 bill. How much change do ou receive? 11

36 0 180 Name Date Chapter Cumulative Review (continued) In Eercises 37 6, use the diagram to find the angle measure. 37. m AOC 38. m AOD 39. m BOE 0. m AOE 1. m COD. m EOD C D E m COE. m AOB m COB 6. m BOD A O B In Eercises 7 5, find the area of the triangle ft 33 d 35 mm 1 mm 13 ft d cm in. 5 cm 18 in. 30 mi 9 mi 53. The length and width of a tissue bo are.5 inches, and the height is 5 inches. What is the volume of the tissue bo? 5. The length of a cereal bo is 7 5 inches, the width is 3 inches, and the height 8 is 11 inches. What is the volume of the cereal bo? 55. A local discount warehouse store is running a special on famil-size cans of soup. The cost for 1 famil-size cans of soup is $5. a. How much is one can of soup? b. Each famil-size can of soup is 50 ounces. What is the price of soup per ounce? Round our answer to the nearest cent. 1

37 Name Date Chapter Cumulative Review (continued) In Eercises 56 67, write an equation of the line that passes through the given point and has the given slope. 56. ( 11, 9 ); m = ( 3, 5 ); m = 58. ( 10, 1 1 ); m = 59. ( 1, 6 ); m = (, 10 1 ); m = 61. ( 9, 3 ); m = 6. ( 1, 8 1 ); m = 63. ( 8, 5 3 ); m = 6. ( 7, 5 ); m = 65. (, 1 1 ); m = 66. ( 5, 7 ); m = ( 9, 0 1 ); m = In Eercises 68 73, write an equation of the line passing through point P that is parallel to the given line. P 3, ; ( ) = 69. P( 0, 7 ); = 1 ( + 6) 70. P 1 (, 5 ); = ( ) 8 P, 0 ; = 7 7. P 1 ( 3, 1 ); = 73. ( ) 3 P, 8 ; 3 + = 1 In Eercises 7 79, write an equation of the line passing through point P that is perpendicular to the given line. P, ; 7 7. ( ) = ( ) P 1 (, 3 ); = ( ) 78. P( 0, 5 ); = 79. ( ) P, 8 ; = + 6 P 5, 1 ; = 5 1 P, 6 ; = 10 are ( ) ( ) ( ) In Eercises 80 87, the vertices of ABC A 1, 3, B, 6, C 0,. Find the coordinates of the vertices of the image after the translation. 80. (, ) ( +, 3) 81. (, ) ( +, + 3) 8. (, ) (, + 1) 83. (, ) ( + 6, 1) 8. (, ) ( + 1, ) 85. (, ) ( + 3, 5) 86. (, ) ( +, + 5) 87. (, ) ( 3, + )