Lesson 9-6 Dilations 587

Objective To understand dilation images of figures

9-6 Dilations

Do you think you can model this using rigid motions?

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.

Normal Light

Diameter of pupil = 2 mm Diameter of pupil = 8 mm

Dim Light

12 mm 12 mm

Iris Pupil

Normal Light

Diameter of pupil = 2 mm

Iris Pupil

Diameter of pupil = 8 mm

Dim Light

12 mm 12 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

LessonVocabulary

A dilation with center of dilation C and scale factor n, n 7 0, can be written as D(n, C). A dilation is a transformation with the following properties.• TheimageofC is itself (that is, C′ = C).

• ForanyotherpointR, R′ is on CR> and

CR′ = n # CR, or n = CR′CR .

• Dilationspreserveanglemeasure.

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C� � C

P�

PQ�Q

R�R

CR� � n � CRS

IntheSolveIt,youlookedathowthepupilofaneyechangesinsize,ordilates. In this lesson,youwilllearnhowtodilategeometricfigures.

Essential Understanding Youcanuseascalefactortomakealargerorsmallercopyofafigurethatisalsosimilartotheoriginalfigure.

Key Concept Dilation

G-SRT.A.1a A dilation takes a line not passing through the center of the dilation to a parallel line, . . . Also G-SRT.A.1b, G-CO.A.2, G-SRT.A.2

MP 1, MP 3, MP 4, MP 7

Common Core State Standards

MATHEMATICAL PRACTICES

Problem 1

588 Chapter 9 Transformations

Thescalefactorn of a dilation is the ratio of a length of the image to the corresponding lengthinthepreimage,withtheimagelengthalwaysinthenumerator.Forthefigure

shown on page 587, n = CR′CR = R′P′

RP = P′Q′PQ = Q′R′

QR .

A dilation is an enlargement if the scale factor nisgreaterthan1.Thedilationisareduction if the scale factor n is between 0 and 1.

Finding a Scale Factor

Multiple Choice Is D(n, X)(△XTR) = △X′T′R′ an enlargement or a reduction? What is the scale factor n of the dilation?

enlargement; n = 2 reduction; n = 13

enlargement; n = 3 reduction; n = 3

Theimageislargerthanthepreimage,sothedilationisanenlargement. Use the ratio of the lengths of corresponding sides to find the scale factor.

n = X′T ′XT = 4 + 8

4 = 124 = 3

△X ′T ′R′ is an enlargement of △XTR,withascalefactorof3.ThecorrectanswerisB.

1. Is D(n, O) (JKLM) = J′K′L′M′an enlargement or a reduction? What is the scale factor n of the dilation?

InGotIt1,youlookedatadilationofafiguredrawninthecoordinateplane.Inthisbook,alldilationsoffiguresinthecoordinateplanehavetheoriginasthecenterofdilation.So youcanfindthedilationimageofapointP(x, y)bymultiplyingthe coordinates of Pbythescalefactorn. A dilation of scale factor n with center of dilation at the origin can be written as

Dn (x, y) = (nx, ny)

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Why is the scale factor not 4

12, or 13?

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.

1.75 in.1.75 in.

Problem 2

Problem 3

Lesson 9-6 Dilations 589

Finding a Dilation Image

What are the coordinates of the vertices of D2(△PZG)? Graph the image of △PZG.

Identifythecoordinatesofeachvertex.Thecenterofdilation is the origin and the scale factor is 2, so use the dilation rule D2(x, y) = (2x, 2y).

D2(P) = (2 # 2, 2 # (-1)), or P′(4, -2).

D2(Z) = (2 # (-2), 2 # 1), or Z′(-4, 2).

D2(G) = (2 # 0, 2 # (-2)), or G′(0, -4).

To graph the image of △PZG, graph P′, Z′, and G′. Thendraw△P′Z′G′.

2. a. WhatarethecoordinatesoftheverticesofD12

(△PZG)?

b. Reasoning How are PZ and P′Z′related? How are PG and P′G′, and GZ

and G′Z′related?Usetheserelationshipstomakeaconjectureabouttheeffects of dilations on lines.

Dilationsandscalefactorshelpyouunderstandreal-worldenlargementsandreductions, such as images seen through a microscope or on a computer screen.

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?

1.75 = 7 # p image length = scale factor # actual length

0.25 = p Divide each side by 7.

Theactuallengthoftheappleseedis0.25in.

3. Theheightofadocumentonyourcomputerscreenis20.4cm.Whenyouchangethezoomsettingonyourscreenfrom100%to25%,thenewimageofyourdocumentisadilationofthepreviousimagewithscalefactor0.25.Whatisthe height of the new image?

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Got It?

Got It?

Will the vertices of the triangle move closer to (0, 0) or farther from (0, 0)?The scale factor is 2, so the dilation is an enlargement. The vertices will move farther from (0, 0).

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 lengthactual length = 7.

Lesson Check

590 Chapter 9 Transformations

Practice and Problem-Solving Exercises

The blue figure is a dilation image of the black figure. The labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. Then find the scale factor of the dilation.

7. 8. 9.

10. 11. 12.

13. 14. 15.

PracticeA See Problem 1.

A 64

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Do you know HOW? 1. Thebluefigureisadilation

imageoftheblackfigurewithcenter of dilation C. Is the dilation an enlargement or a reduction? What is the scale factor of the dilation?

Find the image of each point.

2. D2 (1, -5) 3. D12 (0, 6) 4. D10 (0, 0)

Do you UNDERSTAND? 5. Vocabulary Describethescalefactorofareduction.

6. Error Analysis Thebluefigure is a dilation image oftheblackfigureforadilation with center A.

Twostudentsmadeerrorswhenaskedtofindthescalefactor.Explainandcorrecttheirerrors.

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13=n =

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MATHEMATICAL PRACTICES

MATHEMATICAL PRACTICES

Lesson 9-6 Dilations 591

Find the images of the vertices of △PQR for each dilation. Graph the image.

16. D3 (△PQR) 17. D10 (△PQR) 18. D34 (△PQR)

Magnification You look at each object described in Exercises 19–22 under a magnifying glass. Find the actual dimension of each object.

19. Theimageofabuttonis5timesthebutton’sactualsizeandhasadiameterof6cm.

20. Theimageofapinheadis8timesthepinhead’sactualsizeandhasawidthof1.36 cm.

21. Theimageofanantis7timestheant’sactualsizeandhasalengthof1.4cm.

22. TheimageofacapitalletterNis6timestheletter’sactualsizeandhasaheightof1.68 cm.

Find the image of each point for the given scale factor.

23. L(-3, 0); D5 (L) 24. N(-4, 7); D0.2 (N) 25. A(-6, 2); D1.5 (A)

26. F(3, -2); D13 (F) 27. B(5

4, -32 ); D 1

10 (B) 28. Q(6, 13

2 ); D16 (Q)

Use the graph at the right. Find the vertices of the image of QRTW for a dilation with center (0, 0) and the given scale factor.

29. 14 30. 0.6 31. 0.9 32. 10 33. 100

34. Compare and Contrast Compare the definition of scale factor of a dilation tothedefinitionofscalefactoroftwosimilarpolygons.Howaretheyalike?Howaretheydifferent?

35. Think About a Plan Thediagramattherightshows△LMN and its image △L′M′N′ for a dilation with center P.Findthevaluesofx and y.Explainyourreasoning.

• What is the relationship between △LMN and △L′M′N′? • What is the scale factor of the dilation? • Whichvariablecanyoufindusingthescalefactor?

36. Writing Anequilateraltrianglehas4-in.sides.Describeitsimageforadilationwithcenteratoneofthetriangle’sverticesandscalefactor2.5.

See Problem 2.

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592 Chapter 9 Transformations

Coordinate Geometry Graph MNPQ and its image M′N′P′Q′ 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 38. M(2, 6), N(-4, 10), P(-4, -8), Q(-2, -12); 14

39. Open-Ended Usethedilationcommandingeometrysoftwareordrawingsoftwaretocreateadesignthatinvolvesrepeateddilations,suchastheoneshownattheright.Thesoftwarewillpromptyoutospecifyacenterofdilationandascalefactor.Printyourdesignandcolorit.Feelfreetouseothertransformationsalongwithdilations.

A dilation maps △HIJ onto △H′I′J′. Find the missing values.

40. HI = 8 in. H′I′ = 16 in. 41. HI = ■ ft H′I′ = 8 ft

IJ = 5 in. I′J′ = ■ in. IJ = 30 ft I′J′ = ■ ft

HJ = 6 in. H′J′ = ■ in. HJ = 24 ft H′J′ = 6 ft

42. Let / be a line through the origin. Show that Dk(/) = /byshowingthatifC = (c1, c2) is on /, then Dk(C) is also on /.

43. Let A = (a1, a2) and B = (b1, b2), let A′ = Dk(A) and B′ = Dk(B) with k ≠ 1, and

suppose that <AB

> does not pass through the origin.

a. Show that <AB

>≠

<A′B′

> (Hint: What happens to the x-andy-interceptsof

<AB

>

under the dilation Dk?)

b. Suppose that a1 ≠ b1. Show that <AB

> is parallel to

<A′B′

>byshowingthatthey

havethesameslope.

c. Show that <AB

>� �<A′B′

> if a1 = b1.

44. Reasoning YouaregivenAB and its dilation image A′B′ with A, B, A′, and B′ noncollinear.Explainhowtofindthecenterofdilationandscalefactor.

Reasoning Write true or false for Exercises 45–48. Explain your answers.

45. Adilationisanisometry.

46. A dilation with a scale factor greater than 1 is a reduction.

47. Foradilation,correspondinganglesoftheimageandpreimagearecongruent.

48. Adilationimagecannothaveanypointsincommonwithitspreimage.

Coordinate Geometry In the coordinate plane, you can extend dilations to include scale factors that are negative numbers. For Exercises 49 and 50, use △PQR with vertices P(1, 2), Q(3, 4), and R(4, 1).

49. Graph D-3 (△PQR).

50. a. Graph D-1 (△PQR). b. Explainwhythedilationinpart(a)maybecalledareflection through a point.

Extendyourexplanationtoanewdefinitionofpointsymmetry.

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ChallengeC

Lesson 9-6 Dilations 593

51. Shadows Aflashlightprojectsanimage of rectangle ABCD on a wall so that each vertexofABCDis3 ftawayfromthecorrespondingvertexofA′B′C′D′.Thelength of ABis3 in.ThelengthofA′B′ is 1 ft.HowfarfromeachvertexofABCD is the light?

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Mixed Review 56. △JKLhasverticesJ(23, 2), K(4, 1), and L(1, 23). What are the coordinates of

J ′, K′, and L′ if (Rx@axis ∘ T62, -37)(△JKL) = △J ′K ′L′?

Get Ready! To prepare for Lesson 9-7, do Exercises 55–57.

Algebra TRSU ∼ NMYZ. Find the value of each variable.

57. a 58. b 59. c

See Lesson 9-5.

See Lesson 7-2.

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Standardized Test Prep

52. A dilation maps △CDE onto △C′D′E′. If CD = 7.5 ft, CE = 15 ft, D′E′ = 3.25 ft, and C′D′ = 2.5 ft, what is DE?

1.08 ft 5 ft 9.75 ft 19 ft

53. Youwanttoproveindirectlythatthediagonalsofarectanglearecongruent.Asthefirststepofyourproof,whatshouldyouassume?

A quadrilateral is not a rectangle.

Thediagonalsofarectanglearenotcongruent.

A quadrilateral has no diagonals.

Thediagonalsofarectanglearecongruent.

54. Whichwordcandescribeakite?

equilateral equiangular convex scalene

55. Use the figure at the right to answer the questions below. a. Doesthefigurehaverotationalsymmetry?Ifso,identifytheangleofrotation. b. Doesthefigurehavereflectionalsymmetry?Ifso,howmanylinesofsymmetry

doesithave?

SAT/ACT

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ShortResponse