-16
CC
9-6
Content Standards
Dilations
Objective
G.SRT.1a 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.SRT.1b, G.CO.2, G.SRT.2
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?
MATHEMATICAL
PRACTICES
Normal Light
Dim Light
Iris
Pupil
12 mm
12 mm
Diameter of pupil = 2 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
tdilation
tdilation
tcenter of dilation
tscale factor of a
dilation
tenlargement
treduction
*OUIF4PMWF*UZPVMPPLFEBUIPXUIFQVQJMPGBOFZFDIBOHFTJOTJ[FPSdilates. In this
MFTTPOZPVXJMMMFBSOIPXUPEJMBUFHFPNFUSJDêHVSFT
Essential Understanding :PVDBOVTFBTDBMFGBDUPSUPNBLFBMBSHFSPSTNBMMFS
DPQZPGBêHVSFUIBUJTBMTPTJNJMBSUPUIFPSJHJOBMêHVSF
Key Concept Dilation
P
A dilation with center of dilation C and scale
factor n, n 0, can be written as D(n, C). A dilation
is a transformation with the following properties.
P
r ǔFJNBHFPGC is itself (that is, CR C).
Q
r 'PSBOZPUIFSQPJOUR, RR is on CR and
CRR n CR, or n CRR
CR .
R
R
CC
r %JMBUJPOTQSFTFSWFBOHMFNFBTVSF
8
Q
CRn CR
Lesson 9-6
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Dilations
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ǔFTDBMFGBDUPSn of a dilation is the ratio of a length of the image to the corresponding
MFOHUIJOUIFQSFJNBHFXJUIUIFJNBHFMFOHUIBMXBZTJOUIFOVNFSBUPS'PSUIFêHVSF
CRR
RRPR
PRQR
QRRR
shown on page 587, n CR RP PQ QR .
A dilation is an enlargement if the scale factor nJTHSFBUFSUIBOǔFEJMBUJPOJTB
reduction if the scale factor n is between 0 and 1.
B
B
C
D
D
E
Enlargement
center A, scale factor 2
Problem 1
G
F
C
E
4
2
A A
F
C
2
G
H
Reduction
center C, scale factor 1
4
Finding a Scale Factor
hsm11gmse_0905_t08140.ai
Multiple Choice Is D(n, X)(GXTR) GXRTRRR an enlargement
or a reduction? What is the scale factor n of the dilation?
enlargement; n 2
Why is the scale
4
1
12 , or 3 ?
factor not
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.
H
8
enlargement; n 3
reduction; n 13
reduction; n 3
T
R
8
R
T
X X
ǔF
ǔFJNBHFJTMBSHFSUIBOUIFQSFJNBHFTPUIFEJMBUJPOJTBO
enlargement.
Use the ratio of the lengths of corresponding sides to find the scale factor.
XRTR
4
48
n XT 4 12
hsm11gmse_0905_t08144.ai
4 3
NX RT RRR is an enlargement of NXTRXJUIBTDBMFGBDUPSPGǔFDPSSFDUBOTXFSJT#
y J
Got It? 1. Is D(n, O) (JKLM) JRKRLRMRan enlargement or a
J
reduction? What is the scale factor n of the dilation?
2
2
O
M
M
3
*O(PU*UZPVMPPLFEBUBEJMBUJPOPGBêHVSFESBXOJOUIF
DPPSEJOBUFQMBOF*OUIJTCPPLBMMEJMBUJPOTPGêHVSFTJOUIF
DPPSEJOBUFQMBOFIBWFUIFPSJHJOBTUIFDFOUFSPGEJMBUJPO4P
ZPVDBOêOEUIFEJMBUJPOJNBHFPGBQPJOUP(x, y
CZNVMUJQMZJOH
the coordinates of PCZUIFTDBMFGBDUPSn. A dilation of scale factor
n with center of dilation at the origin can be written as
Dn (x, y) (nx, ny)
588
2
Chapter 9
Transformations
K
K
4
x
L
L
hsm11gmse_0905_t08145.ai
y
P(nx, ny)
ny
P(x, y)
y
OP n OP
x
O
x
nx
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Common Core
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Problem 2
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).
Finding a Dilation Image
2
Z
What are the coordinates of the vertices of
D2 (GPZG)? Graph the image of GPZG.
y
x
O
6 4 2
*EFOUJGZUIFDPPSEJOBUFTPGFBDIWFSUFYǔFDFOUFSPG
*EFOUJGZ
dilation is the origin and the scale factor is 2, so use the
dilation rule D2(x, y) (2x, 2y).
P
4
G
Z
D2(P) (2 2, 2 (1)), or PR(4, 2).
2
Z
y
P x
geom12_se_ccs_c09l06_t03.ai
O
P
4
6 4
D2(Z) (2 (2), 2 1), or ZR(4, 2).
D2(G) (2 0, 2 (2)), or GR(0, 4).
G
To graph the image of NPZG, graph PR, ZR, and GR.
ǔFOESBX NPRZRGR.
G
Got It? 2. a. 8IBUBSFUIFDPPSEJOBUFTPGUIFWFSUJDFTPGD1 (NPZG)?
2
b. Reasoning How are PZ and PRZRrelated? How are PG and PRGR, and GZ
and GRZRSFMBUFE 6TFUIFTFSFMBUJPOTIJQTUPNBLFBDPOKFDUVSFBCPVUUIF
geom12_se_ccs_c09l06_t04.ai
effects of dilations on lines.
%JMBUJPOTBOETDBMFGBDUPSTIFMQZPVVOEFSTUBOESFBMXPSMEFOMBSHFNFOUTBOE
reductions, such as images seen through a microscope or on a computer
screen.
Problem 3
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
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.
ǔFBDUVBMMFOHUIPGUIFBQQMFTFFEJTJO
ǔF
Got It? 3. ǔFIFJHIUPGBEPDVNFOUPOZPVSDPNQVUFS
1.75 in.
TDSFFOJTDN8IFOZPVDIBOHFUIF[PPN
TFUUJOHPOZPVSTDSFFOGSPNUPUIF
OFXJNBHFPGZPVSEPDVNFOUJTBEJMBUJPOPGUIF
QSFWJPVTJNBHFXJUITDBMFGBDUPS8IBUJT
the height of the new image?
Lesson 9-6 Dilations
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Lesson Check
Do you know HOW?
1. ǔFCMVFêHVSFJTBEJMBUJPO
JNBHFPGUIFCMBDLêHVSFXJUI
center 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. D1 (0, 6)
2
Practice
PRACTICES
12
5. Vocabulary %FTDSJCFUIFTDBMFGBDUPSPGBSFEVDUJPO
8
6. Error Analysis ǔFCMVF
figure is a dilation image
PGUIFCMBDLêHVSFGPSB
dilation with center A.
C
2
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4. D10 (0, 0)
A.
3
B.
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MATHEMATICAL
hsm11gmse_0905_t09442.ai
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PRACTICES
8.
A
9.
3
6
hsm11gmse_0905_t06790.ai
R
11.
12.
hsm11gmse_0905_t06792.ai
hsm11gmse_0905_t06791.ai
K
13.
2
4
9
10.
See Problem 1.
C
4
1
6
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.
A
5XPTUVEFOUTNBEFFSSPSTXIFOBTLFEUPêOEUIF
TDBMFGBDUPS&YQMBJOBOEDPSSFDUUIFJSFSSPST
Practice and Problem-Solving Exercises
A
MATHEMATICAL
Do you UNDERSTAND?
M
L
14.
y
6
hsm11gmse_0905_t06793.ai
4
y
15.
y
6
1
x
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O
2
3
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4
2
x
x
O
590
Chapter 9
2
4
6
O
Transformations
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4
6 4 2
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Common Core
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See Problem 2.
Find the images of the vertices of GPQR for each dilation. Graph the image.
16. D3 (NPQR)
4
17. D10 (NPQR)
2
R
P
Q y
3
x
O
P
1
4
5
1
x
O
x
1
y
2
P
2
O
4
Q
Q
y
18. D 3 (NPQR)
2
3
R
R
See Problem 3.
Magnification You look at each object described in Exercises 19–22 under a
magnifying glass. Find the actual dimension of each object.
hsm11gmse_0905_t06800.ai
hsm11gmse_0905_t06801.ai
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19. ǔF
JNBHF PG B CVUUPO JT UJNFT UIF CVUUPOT
BDUVBM TJ[F BOE IBT B EJBNFUFS
PG DN
20. ǔF JNBHF PG B QJOIFBE JT UJNFT UIF QJOIFBET BDUVBM TJ[F BOE IBT B XJEUI PG
1.36 cm.
21. ǔF JNBHF PG BO BOU JT UJNFT UIF BOUT BDUVBM TJ[F BOE IBT B MFOHUI PG DN
22. ǔF JNBHF PG B DBQJUBM MFUUFS / JT UJNFT UIF MFUUFST BDUVBM TJ[F BOE IBT B IFJHIU PG
1.68 cm.
B
Apply
Find the image of each point for the given scale factor.
23. L(3, 0); D5 (L)
24. N(4, 7); D0.2 (N)
26. F(3, 2); D1 (F)
27. B + 4, 2 , ; D 1 (B)
10
3
5
3
25. A(6, 2); D1.5 (A)
3
28. Q + 6, 2 , ; D6 (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
Q
y
W
4
33. 100
2
T
34. Compare and Contrast Compare the definition of scale factor of a dilation
UPUIFEFêOJUJPOPGTDBMFGBDUPSPGUXPTJNJMBSQPMZHPOT)PXBSFUIFZBMJLF )PXBSFUIFZEJŀFSFOU
35. Think About a Plan ǔFEJBHSBNBUUIFSJHIUTIPXTNLMN and
its image NLRMRNR for a dilation with center P'JOEUIFWBMVFTPG
x and y&YQMBJOZPVSSFBTPOJOH
t What is the relationship between NLMN and NLRMRNR?
t What is the scale factor of the dilation?
t 8IJDIWBSJBCMFDBOZPVêOEVTJOHUIFTDBMFGBDUPS
3
1
R
L
4
O
2
x
x3
hsm11gmse_0905_t06802.ai
M
L
2
P
x
M
y N
N
(2y 60)
36. Writing "OFRVJMBUFSBMUSJBOHMFIBTJOTJEFT%FTDSJCFJUTJNBHFGPS
BEJMBUJPOXJUIDFOUFSBUPOFPGUIFUSJBOHMFTWFSUJDFTBOETDBMFGBDUPS
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Lesson 9-6 Dilations
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Coordinate Geometry Graph MNPQ and its image MNPQ 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 6TFUIFEJMBUJPODPNNBOEJOHFPNFUSZTPGUXBSFPS
ESBXJOHTPGUXBSFUPDSFBUFBEFTJHOUIBUJOWPMWFTSFQFBUFEEJMBUJPOT
TVDIBTUIFPOFTIPXOBUUIFSJHIUǔFTPGUXBSFXJMMQSPNQUZPVUP
TQFDJGZBDFOUFSPGEJMBUJPOBOEBTDBMFGBDUPS1SJOUZPVSEFTJHOBOE
DPMPSJU'FFMGSFFUPVTFPUIFSUSBOTGPSNBUJPOTBMPOHXJUIEJMBUJPOT
40. Copy Reduction :PVSQJDUVSFPGZPVSGBNJMZDSFTUJTJOXJEF:PV
OFFEBSFEVDFEDPQZGPSUIFGSPOUQBHFPGUIFGBNJMZOFXTMFUUFSǔF
DPQZNVTUêUJOBTQBDFJOXJEF8IBUTDBMFGBDUPSTIPVMEZPVVTF
POUIFDPQZNBDIJOFUPBEKVTUUIFTJ[FPGZPVSQJDUVSFPGUIFDSFTU
A dilation maps GHIJ onto GHRIRJR. Find the missing values.
41. HI 8 in.
HRIR 16 in.
42. HI 7 cm
HRIR 5.25 cm 43. HI J ft
HRIR 8 ft
IJ 5 in.
IRJR J in.
IJ 7 cm
IRJR J cm
IJ hsm11gmse_0905_t09542.ai
30 ft
IRJR J ft
HJ 6 in.
HRJR J in.
HJ J cm
HRJR 9 cm
HJ 24 ft
T
Copy GTBA and point O for each of Exercises 44–47. Draw the dilation
image GTBA.
44. D(2, O) (NTBA)
45. D(3, B) (NTBA)
46. D(1, T) (NTBA)
3
47. D(1, O) (NTBA)
2
HRJR 6 ft
B
O
A
48. Reasoning :PVBSFHJWFOAB and its dilation image ARBR with A, B, AR, and BR
OPODPMMJOFBS&YQMBJOIPXUPêOEUIFDFOUFSPGEJMBUJPOBOETDBMFGBDUPS
Reasoning Write true or false for Exercises 49–52. Explain your answers.
hsm11gmse_0905_t06805.ai
49. "EJMBUJPOJTBOJTPNFUSZ
50. A dilation with a scale factor greater than 1 is a reduction.
51. 'PSBEJMBUJPODPSSFTQPOEJOHBOHMFTPGUIFJNBHFBOEQSFJNBHFBSFDPOHSVFOU
52. "EJMBUJPOJNBHFDBOOPUIBWFBOZQPJOUTJODPNNPOXJUIJUTQSFJNBHF
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
GPQR with vertices P(1, 2), Q(3, 4), and R(4, 1).
53. Graph D3 (NPQR).
54. a. Graph D1 (NPQR).
b. &YQMBJOXIZUIFEJMBUJPOJOQBSUB
NBZCFDBMMFEBreflection through a point.
&YUFOEZPVSFYQMBOBUJPOUPBOFXEFêOJUJPOPGQPJOUTZNNFUSZ
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Chapter 9
Transformations
Common Core
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55. Shadows "ëBTIMJHIUQSPKFDUTBOJNBHF
of rectangle ABCD on a wall so that each
WFSUFYPGABCDJTGUBXBZGSPNUIF
DPSSFTQPOEJOHWFSUFYPGARBRCRDRǔF
length of ABJTJOǔFMFOHUIPGARBR is
GU)PXGBSGSPNFBDIWFSUFYPGABCD
is the light?
B
C
B
C
A D
A
D
Standardized Test Prep
56. A dilation maps NCDE onto NCRDRER. If CDhsm11gmse_0905_t06807.ai
7.5 ft, CE 15 ft, DRER 3.25 ft,
and CRDR 2.5 ft, what is DE?
SAT/ACT
1.08 ft
5 ft
9.75 ft
19 ft
57. :PVXBOUUPQSPWFJOEJSFDUMZUIBUUIFEJBHPOBMTPGBSFDUBOHMFBSFDPOHSVFOU"TUIF
êSTUTUFQPGZPVSQSPPGXIBUTIPVMEZPVBTTVNF
A quadrilateral is not a rectangle.
ǔFEJBHPOBMTPGBSFDUBOHMFBSFOPUDPOHSVFOU
A quadrilateral has no diagonals.
ǔFEJBHPOBMTPGBSFDUBOHMFBSFDPOHSVFOU
58. 8IJDIXPSEDBOEFTDSJCFBLJUF
equilateral
equiangular
DPOWFY
scalene
59. Use the figure at the right to answer the questions below.
a. %PFTUIFêHVSFIBWFSPUBUJPOBMTZNNFUSZ *GTPJEFOUJGZUIFBOHMFPGSPUBUJPO
b. %PFTUIFêHVSFIBWFSFëFDUJPOBMTZNNFUSZ *GTPIPXNBOZMJOFTPGTZNNFUSZ
EPFTJUIBWF
Short
Response
Mixed Review
hsm11gmse_0905_t09443.ai
See
Lesson 9-5.
60. NJKLIBTWFSUJDFT J(23, 2), K(4, 1), and L(1, 23). What are the coordinates of
J R, KR, and LR if (RxaYis T2, 3)(NJKL) NJ RK RLR?
Get Ready! To prepare for Lesson 9-7, do Exercises 61–63.
Algebra TRSU NMYZ. Find the value of each variable.
61. a
62. b
See Lesson 7-2.
M
63. c
R
4.5
3
N
T 120
a
S
50
150 Y
b
6
5
c
Z
U
Lesson 9-6 Dilations
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Dilations
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