Key Concept Dilation

Transcription

1 Dilations ontent Standards G.ST.a 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. Also G.ST.b, G.., G.ST. bjective 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 bserve 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 d center of dilation scale factor of a dilation enlargement reduction dilates. In this Essential Understanding Key oncept Dilation A dilation with center of dilation and scale factor n, n 0, can be written as D (n, ). A dilation is a transformation with the following properties. is itself (that is, ). Q, is on and n, or n. 8 n Q -6 Dilations

2 n of a dilation is the ratio of a length of the image to the corresponding shown on page 587, n Q Q Q Q. A dilation is an enlargement if the scale factor n reduction if the scale factor n is between 0 and. 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) XT an enlargement or a reduction? What is the scale factor n of the dilation? enlargement; n reduction; n enlargement; n reduction; n X X enlargement. Use the ratio of the lengths of corresponding sides to find the scale factor. n XT XT 8 X T is an enlargement of XT T 8 T Got It?. Is D (n, ) (JKLM) JKLMan enlargement or a reduction? What is the scale factor n of the dilation? y J J K M M L K L (, y the coordinates of 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 y (n, ny) (, y) n n ommon ore

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. 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. ZG. y Z 6 G Z Z y 6 G G Got It?. a. D (ZG)? b. easoning How are Z and Zrelated? How are G and G, and GZ and GZ effects of dilations on lines. 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 7. actual length roblem Using a Scale Factor to Find a Length iology 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. Got It?. the height of the new image?.75 in. -6 Dilations

4 Lesson heck Do you know HW?. 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, 6). D 0 (0, 0) 8 Do you UNDESTAND? 5. Vocabulary 6. Error Analysis figure is a dilation image dilation with center A. A.. MATHEMATIAL ATIES 6 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 ommon ore

5 Find the images of the vertices of Q for each dilation. Graph the image. 6. D (Q) 7. D 0 (Q) 8. D (Q) y Q Q y Q y 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. Apply cm cm. Find the image of each point for the given scale factor.. L(, 0); D 5 (L). N(, 7); D 0. (N) 5. A(6, ); D.5 (A) 6. F(, ); D (F) 7. 5, ; D () 0 8. Q6, ; D 6 (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 5. Think About a lan LMN and its image LMN for a dilation with center and y What is the relationship between LMN and LMN? What is the scale factor of the dilation? 6. Writing y W Q T L y L M M N N (y 60) -6 Dilations 5

6 oordinate Geometry Graph MNQ and its image MNQ for a dilation with center (0, 0) and the given scale factor. 7. M(, ), N(, ), (5, ), Q(, ); 8. M(, 6), N(, 0), (, 8), Q(, ); 9. pen-ended 0. opy eduction A dilation maps HIJ onto HIJ. Find the missing values.. HI 8 in. HI 6 in.. HI 7 cm HI 5.5 cm. HI ft HI 8 ft IJ 5 in. IJ in. IJ 7 cm IJ cm IJ 0 ft IJ ft HJ 6 in. HJ in. HJ cm HJ 9 cm HJ ft HJ 6 ft opy TA and point for each of Eercises 7. Draw the dilation image TA. T. D (, ) (TA) 5. D (, ) (TA) 6. D (, T) (TA) 7. D (, ) (TA) 8. easoning A and its dilation image A with A,, A, and easoning Write true or false for Eercises 9 5. Eplain your answers A dilation with a scale factor greater than is a reduction A hallenge oordinate Geometry In the coordinate plane, you can etend dilations to include scale factors that are negative numbers. For Eercises 5 and 5, use Q with vertices (, ), Q(, ), and (, ). 5. Graph D (Q). 5. a. Graph D (Q). b. reflection through a point. 6 ommon ore

7 55. Shadows of rectangle AD on a wall so that each AD AD length of AA is AD is the light? A D A D Standardized Test rep SAT/AT 56. A dilation maps DE onto DE. If D 7.5 ft, E 5 ft, DE.5 ft, and D.5 ft, what is DE?.08 ft 5 ft 9.75 ft 9 ft 57. A quadrilateral is not a rectangle. A quadrilateral has no diagonals. 58. equilateral equiangular scalene Short esponse 59. Use the figure at the right to answer the questions below. a. b. Mied eview 60. JKLJ(, ), K(, ), and L(, ). What are the coordinates of J, K, and L if ( ais T, )(JKL) J K L? See Lesson 9-5. Get eady! To prepare for Lesson 9-7, do Eercises 6 6. Algebra TSU NMYZ. Find the value of each variable. 6. a 6. b 6. c T 0 a 5 S M.5 50 N 50 Y 6 b c See Lesson 7-. U Z -6 Dilations 7