Coordinate Geometry Graph MNPQ and its image M ′ N ′ P ′ Q ′ m prime , n prime , p prime , q prime  for a dilation with center (0, 0) and the given scale factor.

37. M ( 1 , 3 ) , N ( − 3 , 3 ) , P ( − 5 , − 3 ) , Q ( − 1 , − 3 ) ; 3 m open 1 comma 3 close comma n open negative 3 comma 3 close comma p open negative 5 comma negative 3 close comma q open negative 1 comma negative 3 close semicolon 3
38. M ( 2 , 6 ) , N ( − 4 , 10 ) , P ( − 4 , − 8 ) , Q ( − 2 , − 12 ) ; 1 4 m open 2 comma 6 close comma n open negative 4 comma 10 close comma p open negative 4 comma negative 8 close comma q open negative 2 comma negative 12 close semicolon , 1 fourth

39. 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 below. The software will prompt you to specify a center of dilation and a scale factor. Print your design and color it. Feel free to use other transformations along with dilations.

    

40. Copy Reduction Your picture of your family crest is 4.5 in. wide. You need a reduced copy for the front page of the family newsletter. The copy must fit in a space 1.8 in. wide. What scale factor should you use on the copy machine to adjust the size of your picture of the crest?

A dilation maps Δ H I J cap delta h i j  onto Δ H ′ I ′ J ′ . cap delta , h prime , i prime , j prime , .  Find the missing values.

41. H I = 8  in . I J = 5  in . H J = 6  in . H ′ I ′ = 16  in . I ′ J ′ = □  in . H ′ J ′ = □  in . table with 1 row and 2 columns , row1 column 1 , table with 3 rows and 1 column , row1 column 1 , h i equals 8 in . , row2 column 1 , i j equals 5 in . , row3 column 1 , h j equals 6 in . , end table , column 2 table with 3 rows and 1 column , row1 column 1 , h prime , i prime , equals 16 , in . , row2 column 1 , i prime , j prime , equals white square in . , row3 column 1 , h prime , j prime , equals white square in . , end table , end table
42. H I = 7  cm I J = 7  cm H J = □  cm H ′ I ′ = 5.25  cm I ′ J ′ = □  cm H ′ J ′ = 9  cm table with 1 row and 2 columns , row1 column 1 , table with 3 rows and 1 column , row1 column 1 , h i equals 7 cm , row2 column 1 , i j equals 7 cm , row3 column 1 , h j equals white square cm , end table , column 2 table with 3 rows and 1 column , row1 column 1 , h prime , i prime , equals , 5.25 , cm , row2 column 1 , i prime , j prime , equals white square cm , row3 column 1 , h prime , j prime , equals 9 cm , end table , end table
43. H I = □  ft I J = 30  ft H J = 24  ft H ′ I ′ = 8  ft I ′ J ′ = □  ft H ′ J ′ = 6  ft table with 1 row and 2 columns , row1 column 1 , table with 3 rows and 1 column , row1 column 1 , h i equals white square ft , row2 column 1 , i j equals 30 , ft , row3 column 1 , h j equals 24 , ft , end table , column 2 table with 3 rows and 1 column , row1 column 1 , h prime , i prime , equals 8 ft , row2 column 1 , i prime , j prime , equals white square ft , row3 column 1 , h prime , j prime , equals 6 ft , end table , end table

Copy Δ T B A cap delta t b eh  and point O for each of Exercises 44–47. Draw the dilation image Δ T ′ B ′ A ′ cap delta , t prime , b prime , eh prime  for the given center and scale factor.

44. center O, scale factor 2
45. center B, scale factor 3
46. center T, scale factor 1 3 1 third
47. center O, scale factor 1 2 1 half
48. Reasoning You are given A B ¯ eh b bar  and its dilation image A ′ B ′ ¯ eh prime , b prime bar  with A, B, A ′ , eh prime comma  and B ′ b prime  noncollinear. Explain how to find the center of dilation and scale factor.

Reasoning Write true or false for Exercises 49–52. Explain your answers.

49. A dilation is an isometry.
50. A dilation with a scale factor greater than 1 is a reduction.
51. For a dilation, corresponding angles of the image and preimage are congruent.
52. A dilation image cannot have any points in common with its preimage.

C Challenge

Coordinate Geometry In the coordinate plane, you can extend dilations to include scale factors that are negative numbers. For Exercises 53 and 54, use Δ P Q R cap delta p q r  with vertices P(1, 2), Q(3, 4), and R(4, 1).

53. Graph Δ P Q R cap delta p q r  and its image for a dilation centered at (0, 0) with scale factor − 3 . negative 3 .
54. 1. Graph Δ P Q R cap delta p q r  and its image for a dilation centered at (0, 0) with scale factor − 1 . negative 1 .
    2. Explain why the dilation in part (a) may be called a reflection through a point. Extend your explanation to a new definition of point symmetry.

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