Properties Transformations

Transcription

1 9 Properties of Transformations 9. Translate Figures and Use Vectors 9.2 Use Properties of Matrices 9.3 Perform Reflections 9.4 Perform Rotations 9.5 ppl ompositions of Transformations 9.6 Identif Smmetr 9.7 Identif and Perform Dilations efore In previous chapters, ou learned the following skills, which ou ll use in hapter 9: translating, reflecting, and rotating polgons, and using similar triangles. Prerequisite Skills VOULRY HEK Match the transformation of Triangle with its graph.. Translation of Triangle 2. Reflection of Triangle 3. Rotation of Triangle D 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. (Review p. 272 for 9., 9.3.) 4. Translate 3 units left and unit down. 5. Reflect in the -ais. In the diagram, D, EFGH. (Review p. 234 for 9.7.) 6. Find the scale factor of D to EFGH. 7. Find the values of,, and z. z D 0 6 E 2 H 5 F 8 G 570

2 Now In hapter 9, ou will appl the big ideas listed below and reviewed in the hapter Summar on page 635. You will also use the ke vocabular listed below. ig Ideas Performing congruence and similarit transformations 2 Making real-world connections to smmetr and tessellations 3 ppling matrices and vectors in Geometr KEY VOULRY image, p. 572 preimage, p. 572 isometr, p. 573 vector, p. 574 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 rotational smmetr, p. 620 scalar multiplication, p. 627 Wh? You can use properties of shapes to determine whether shapes tessellate. For eample, ou can use angle measurements to determine which shapes can be used to make a tessellation. Geometr The animation illustrated below for Eample 3 on page 67 helps ou answer this question: How can ou use tiles to tessellate a floor? tessellation covers a plane with no gaps or overlaps. hoose tiles and draw a tessellation. You ma translate, reflect, and rotate tiles. Geometr at classzone.com Other animations for hapte r 9 : page s582, 590, 599, 602, 6, 6. 9, and

3 9. Translate Figures 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, p. 272 In Lesson 4.8, ou learned that a 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 P and Q of a plane figure to the points P9 (read P prime ) and Q9, so that one of the following statements is true: P9 PP9 5 QQ9 and } PP9 i } QQ9, or PP9 5 QQ9 and } PP9 and } QQ9 are collinear. P Œ Œ9 E X M P L E Translate a figure in the coordinate plane Graph quadrilateral D with vertices (2, 2), (2, 5), (4, 6), and D(4, 2). Find 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. For eample, if the preimage is n, then its image is n 999, read as triangle prime, prime, prime. Solution First, draw D. Find 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, ) (2, 5) 9(2, 4) (4, 6) 9(7, 5) D(4, 2) D9(7, ) 9 9 D 9 D9 GUIDED PRTIE for Eample. Draw nrst with vertices R(2, 2), S(5, 2), and T(3, 5). Find the image of each verte after the translation (, ) (, 2). Graph the image using prime notation. 2. The image of (, ) ( 4, 2 7) is } P9Q9 with endpoints P9(23, 4) and Q9(2, ). Find the coordinates of the endpoints of the preimage. 572 hapter 9 Properties of Transformations

4 ISOMETRY n isometr is a transformation that preserves length and angle measure. Isometr is another word for congruence transformation (page 272). RED DIGRMS In this book, the preimage is alwas shown in blue, and the image is alwas shown in red. E X M P L E 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 Postulate. 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 PRTIE for Eample 2 3. In Eample 2, write a rule to translate n 999 back to n. THEOREM For Your Notebook THEOREM 9. Translation Theorem translation is an isometr. Proof: below; E. 46, p n > n P RO O F Translation Theorem translation is an isometr. GIVEN c P(a, b) and Q(c, d) are two points on a figure translated b (, ) ( s, t). PROVE c PQ 5 P9Q9 P(a, b) Œ(c, d) P9(a s, b t) Œ9(c s, d t) The translation maps P(a, b) to P9(a s, b t) and Q(c, d) to Q9(c s, d t). Use the Distance Formula to find PQ and P9Q9. PQ 5 Ï }} (c 2 a) 2 (d 2 b) 2. P9Q9 5 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, PQ 5 P9Q9 b the Transitive Propert of Equalit. 9. Translate Figures and Use Vectors 573

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 ONEPT For 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# FG z, read as vector FG. The initial point, or starting point, of the vector is F. horizontal component F 5 units right The component form of a vector combines the horizontal and vertical components. So, the component form of# FG z is 5, 3. G 3 units up The terminal point, or ending point, of the vector is G. vertical component E X M P L E 3 Identif vector components Name the vector and write its component form. a. b. T S Solution a. The vector is# z. From 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. From initial point S to terminal point T, ou move 8 units left and 0 units verticall. The component form is 28, 0. E X M P L E 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 First, 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) 574 hapter 9 Properties of Transformations

6 GUIDED PRTIE 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 nlmn are L(2, 2), M(5, 3), and N(9, ). Translate nlmn using the vector 22, 6. E X M P L E 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 PRTIE for Eample 5 8. WHT IF? 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 Figures and Use Vectors 575

7 9. EXERISES SKILL PRTIE HOMEWORK KEY 5 WORKED-OUT SOLUTIONS on p. WS for Es. 7,, and 35 5 STNDRDIZED TEST PRTIE Es. 2, 4, and 42. VOULRY op and complete:? is a quantit that has both? and magnitude. 2. WRITING Describe the difference between a vector and a ra. EXMPLE on p. 572 for Es. 3 0 IMGE ND PREIMGE 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)? EXMPLE 2 on p. 573 for Es. 4 GRPHING N IMGE The vertices of npqr are P(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 EFGH. (, ) ( 2, 2 2) E H F E H G F G 4. MULTIPLE HOIE Translate Q(0, 28) using (, ) ( 2 3, 2). Q9(22, 5) Q9(3, 20) Q9(23, 26) D Q9(2, 2) EXMPLE 3 on p. 574 for Es IDENTIFYING VETORS Name the vector and write its component form R P D T J 576 hapter 9 Properties of Transformations

8 VETORS Use the point P(23, 6). Find the component form of the vector that describes the translation to P9. 8. P9(0, ) 9. P9(24, 8) 20. P9(22, 0) 2. P9(23, 25) TRNSLTIONS Think of each translation as a vector. Describe the vertical component of the vector. Eplain EXMPLE 4 on p. 574 for Es TRNSLTING TRINGLE The vertices of n DEF are D(2, 5), E(6, 3), and F(4, 0). Translate ndef using the given vector. Graph n DEF and its image , , , , 24 LGER Find 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-STEP PROLEM 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. Find the areas of QRST Q9R9S9T9. b. ompare the areas. Make a conjecture about the areas of a preimage and its image after a translation. 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 DEF, and its image after the transformation given in part (a). re the areas of n DEF and n D9E9F9 the same? 9. Translate Figures and Use Vectors 577

9 PROLEM SOLVING EXMPLE 2 on p. 573 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 F. (0, 4) (3, 4) (0, 0) D E F 34. Write a rule to translate Rectangle F back to Rectangle. (0, 4) (6, 4) EXMPLE 5 on p. 575 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. From the cabin to the ski lodge 36. From the ski lodge to the hotel 37. From the hotel back to our cabin W N E S 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. 40. 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 RESPONSE 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 WORKED-OUT SOLUTIONS on p. WS 5 STNDRDIZED TEST PRTIE

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? D E F G H MULTI-STEP PROLEM 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 Parabola is 5 ( 2 ) 2 3, with verte (, 3). Parabola 2 is the image of Parabola after a translation. a. Write a rule for the translation. b. Write an equation of Parabola 2. c. Suppose ou translate Parabola using the vector 24, 8. Write an equation of the image. d. n equation of Parabola 3 is 5 ( 5) Write a rule for the translation of Parabola to Parabola 3. Eplain our reasoning. (, 3) Parabola Parabola 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 Theorem 9., that a translation preserves angle measure. MIXED REVIEW PREVIEW Prepare for Lesson 9.2 in Es Find the sum, difference, product, or quotient. (p. 869) (22) 49. (3)(22) (24) Determine whether the two triangles are similar. If the are, write a similarit statement. (pp. 38, 388) 5. P D S P R T 20 9 E Points,,, and D are the vertices of a quadrilateral. Give the most specific name for D. Justif our answer. (p. 552) 53. (2, 0), (7, 0), (4, 4), D(2, 4) 54. (3, 0), (7, 2), (3, 4), D(, 2) EXTR PRTIE for Lesson 9., p. 92 ONLINE QUIZ at classzone.com 579