To understand dilation images of figures

Transcription

1 9- Dilations ommon ore State Standards G-ST.A.a A dilation takes a line not passing through the center of the dilation to a parallel line,... Also G-ST.A.b, G-O.A., G-ST.A. M, M, M, M 7 Objective To understand dilation images of figures The pupil is the opening in the iris that lets light into the eye. Depending on the amount of light available, the size of the pupil changes. Do you think you can model this using rigid motions? MATHEMATIAL ATIES Normal Light Dim Light Iris upil mm mm Diameter of pupil = mm Diameter of pupil = 8 mm Observe the size and shape of the iris in normal light and in dim light. What characteristics stay the same and what characteristics change? How do these observations compare to transformation of figures you have learned about earlier in the chapter? Lesson Vocabulary dilation center of dilation scale factor of a dilation enlargement reduction In the Solve It, you looked at how the pupil of an eye changes in size, or dilates. In this lesson, you will learn how to dilate geometric figures. Essential Understanding You can use a scale factor to make a larger or smaller copy of a figure that is also similar to the original figure. Key oncept Dilation A dilation with center of dilation and scale factor n, n 7 0, can be written as D (n, ). A dilation is a transformation with the following properties. The image of is itself (that is, = ). For any other point, is on > and = n #, or n =. Dilations preserve angle measure. Q n S Q Lesson 9- Dilations 587

2 The scale factor n of a dilation is the ratio of a length of the image to the corresponding length in the preimage, with the image length always in the numerator. For the figure shown on page 587, n = = = Q Q = Q Q. A dilation is an enlargement if the scale factor n is greater than. The dilation is a reduction if the scale factor n is between 0 and. B B A A D Enlargement center A, scale factor D E F G F G E H 8 eduction center, scale factor H roblem Finding a Scale Factor Why is the scale factor not, or? The scale factor of a dilation always has the image length (or the distance between a point on the image and the center of dilation) in the numerator. Multiple hoice Is D (n, X) ( XT) = X T an enlargement or a reduction? What is the scale factor n of the dilation? enlargement; n = reduction; n = enlargement; n = reduction; n = The image is larger than the preimage, so the dilation is an X X enlargement. Use the ratio of the lengths of corresponding sides to find the scale factor. n = X T XT = + 8 = = X T is an enlargement of XT, with a scale factor of. The correct answer is B. T 8 T Got It?. Is D (n, O) (JKLM) = J K L M an enlargement or a reduction? What is the scale factor n of the dilation? y J J O K M M L K L In Got It, you looked at a dilation of a figure drawn in the coordinate plane. In this book, all dilations of figures in the coordinate plane have the origin as the center of dilation. So you can find the dilation image of a point (, y) by multiplying the coordinates of by the scale factor n. A dilation of scale factor n with center of dilation at the origin can be written as D n (, y) = (n, ny) ny y O y (n, ny) (, y) O n O n 588 hapter 9 Transformations

3 Will the vertices of the triangle move closer to (0, 0) or farther from (0, 0)? The scale factor is, so the dilation is an enlargement. The vertices will move farther from (0, 0). roblem Finding a Dilation Image What are the coordinates of the vertices of D ( ZG)? Graph the image of ZG. Identify the coordinates of each verte. The center of dilation is the origin and the scale factor is, so use the dilation rule D (, y) = (, y). D () = ( #, # (-)), or (, -). D (Z) = ( # (-), # ), or Z (-, ). D (G) = ( # 0, # (-)), or G (0, -). To graph the image of ZG, graph, Z, and G. Then draw Z G. y Z O G Z Z y O G G Got It?. a. What are the coordinates of the vertices of D ( ZG)? b. easoning How are Z and Z related? How are G and G, and GZ and G Z related? Use these relationships to make a conjecture about the effects of dilations on lines. Dilations and scale factors help you understand real-world enlargements and reductions, such as images seen through a microscope or on a computer screen. What does a scale factor of 7 tell you? A scale factor of 7 tells you that the ratio of the image length to the actual length is 7, or image length actual length = 7. roblem Using a Scale Factor to Find a Length Biology A magnifying glass shows you an image of an object that is 7 times the object s actual size. So the scale factor of the enlargement is 7. The photo shows an apple seed under this magnifying glass. What is the actual length of the apple seed?.75 = 7 # p image length = scale factor # actual length 0.5 = p Divide each side by 7. The actual length of the apple seed is 0.5 in. Got It?. The height of a document on your computer screen is 0. cm. When you change the zoom setting on your screen from 00% to 5%, the new image of your document is a dilation of the previous image with scale factor 0.5. What is the height of the new image?.75 in. Lesson 9- Dilations 589

4 Lesson heck Do you know HOW?. The blue figure is a dilation image of the black figure with center of dilation. Is the dilation an enlargement or a reduction? What is the scale factor of the dilation? Find the image of each point.. D (, -5). D (0, ). D 0 (0, 0) 8 Do you UNDESTAND? 5. Vocabulary Describe the scale factor of a reduction.. Error Analysis The blue figure is a dilation image of the black figure for a dilation with center A. Two students made errors when asked to find the scale factor. Eplain and correct their errors. A. B. n = = MATHEMATIAL ATIES n = = A ractice and roblem-solving Eercises MATHEMATIAL ATIES A ractice The blue figure is a dilation image of the black figure. The labeled point is the See roblem. center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation A K L M. y. y 5. y O O O 590 hapter 9 Transformations

5 Find the images of the vertices of Q for each dilation. Graph the image.. D ( Q) 7. D 0 ( Q) 8. D ( Q) y Q Q y Q y O O O 5 Magnification You look at each object described in Eercises 9 under a magnifying glass. Find the actual dimension of each object. See roblem. See roblem. B Apply 9. The image of a button is 5 times the button s actual size and has a diameter of cm. 0. The image of a pinhead is 8 times the pinhead s actual size and has a width of. cm.. The image of an ant is 7 times the ant s actual size and has a length of. cm.. The image of a capital letter N is times the letter s actual size and has a height of.8 cm. Find the image of each point for the given scale factor.. L(-, 0); D 5 (L). N(-, 7); D 0. (N) 5. A(-, ); D.5 (A). F(, -); D (F) 7. B( 5, - ); D (B) 8. Q (, ); D 0 (Q) Use the graph at the right. Find the vertices of the image of QTW for a dilation with center (0, 0) and the given scale factor ompare and ontrast ompare the definition of scale factor of a dilation to the definition of scale factor of two similar polygons. How are they alike? How are they different? 5. Think About a lan The diagram at the right shows LMN and its image L M N for a dilation with center. Find the values of and y. Eplain your reasoning. What is the relationship between LMN and L M N? What is the scale factor of the dilation? Which variable can you find using the scale factor?. Writing An equilateral triangle has -in. sides. Describe its image for a dilation with center at one of the triangle s vertices and scale factor.5. y W Q T O L y L M M N N (y 0) Lesson 9- Dilations 59

6 oordinate Geometry Graph MNQ and its image M N Q for a dilation with center (0, 0) and the given scale factor. 7. M(, ), N(-, ), (-5, -), Q(-, -); 8. M(, ), N(-, 0), (-, -8), Q(-, -); 9. Open-Ended Use the dilation command in geometry software or drawing software to create a design that involves repeated dilations, such as the one shown at the right. The software will prompt you to specify a center of dilation and a scale factor. rint your design and color it. Feel free to use other transformations along with dilations. A dilation maps HIJ onto H I J. Find the missing values. 0. HI = 8 in. H I = in.. HI = ft H I = 8 ft IJ = 5 in. I J = in. IJ = 0 ft I J = ft HJ = in. H J = in. HJ = ft H J = ft. Let / be a line through the origin. Show that D k (/) = / by showing that if = (c, c ) is on /, then D k () is also on /.. Let A = (a, a ) and B = (b, b ), let A = D k (A) and B = D k (B) with k, and suppose that < AB > does not pass through the origin. a. Show that < AB > < A B > (Hint: What happens to the - and y-intercepts of < AB > under the dilation D k?) b. Suppose that a b. Show that < AB > is parallel to < A B > by showing that they have the same slope. c. Show that < AB > < A B > if a = b.. easoning You are given AB and its dilation image A B with A, B, A, and B noncollinear. Eplain how to find the center of dilation and scale factor. hallenge easoning Write true or false for Eercises 5 8. Eplain your answers. 5. A dilation is an isometry.. A dilation with a scale factor greater than is a reduction. 7. For a dilation, corresponding angles of the image and preimage are congruent. 8. A dilation image cannot have any points in common with its preimage. oordinate Geometry In the coordinate plane, you can etend dilations to include scale factors that are negative numbers. For Eercises 9 and 50, use Q with vertices (, ), Q(, ), and (, ). 9. Graph D - ( Q). 50. a. Graph D - ( Q). b. Eplain why the dilation in part (a) may be called a reflection through a point. Etend your eplanation to a new definition of point symmetry. 59 hapter 9 Transformations

7 5. Shadows A flashlight projects an image of rectangle ABD on a wall so that each verte of ABD is ft away from the corresponding verte of A B D. The length of AB is in. The length of A B is ft. How far from each verte of ABD is the light? B A D A B D Standardized Test rep SAT/AT 5. A dilation maps DE onto D E. If D = 7.5 ft, E = 5 ft, D E =.5 ft, and D =.5 ft, what is DE?.08 ft 5 ft 9.75 ft 9 ft 5. You want to prove indirectly that the diagonals of a rectangle are congruent. As the first step of your proof, what should you assume? A quadrilateral is not a rectangle. The diagonals of a rectangle are not congruent. A quadrilateral has no diagonals. The diagonals of a rectangle are congruent. 5. Which word can describe a kite? equilateral equiangular conve scalene Short esponse 55. Use the figure at the right to answer the questions below. a. Does the figure have rotational symmetry? If so, identify the angle of rotation. b. Does the figure have reflectional symmetry? If so, how many lines of symmetry does it have? Mied eview 5. JKL has vertices J(, ), K(, ), and L(, ). What are the coordinates of J, K, and L if T, -7 )( JKL) = J K L? See Lesson 9-5. Get eady! To prepare for Lesson 9-7, do Eercises Algebra TSU NMYZ. Find the value of each variable. 57. a 58. b 59. c T 0 a 5 S M.5 50 N 50 Y b c See Lesson 7-. U Z Lesson 9- Dilations 59