Embed Size (px)
Citation preview
Lesson 9-6 Dilations 587
Objective To understand dilation images of �gures
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 Vocabularydilationcenter of dilationscale factor of a dilationenlargementreduction
LessonVocabularyVocabularydilationVocabularydilation
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.
C is itself (that is, C C).
R, R is on CR and CR n CR, or n CR
CR .
hsm11gmse_0905_t09541.ai
C C
P
PQQ
RR8
CR n CR
dilates. In this
Essential Understanding
Key Concept Dilation
Content StandardsG.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
MATHEMATICAL PRACTICES
GEOM12_SE_CCS_C09L06.indd 587 7/5/11 2:36:42 PM
CC-16 Dilations 1
CC-16
HSM12GESE_CC16.indd 1 8/1/11 1:26:55 PM
Problem 1
Got It?
588 Chapter 9 Transformations
n of a dilation is the ratio of a length of the image to the corresponding
shown on page 587, n CRCR
R PRP P Q
PQQ RQR .
A dilation is an enlargement if the scale factor nreduction 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
enlargement. Use the ratio of the lengths of corresponding sides to �nd the scale factor.
n X TXT
4 84
124 3
X T R is an enlargement of XTR
1. Is D(n, O) (JKLM) J K L M an enlargement or a reduction? What is the scale factor n of the dilation?
P(x, y
the coordinates of P n. A dilation of scale factor n with center of dilation at the origin can be written as
Dn (x, y) (nx, ny)
hsm11gmse_0905_t08140.ai
Enlargementcenter A, scale factor 2
A A D
B C
D
C 4
8
22
B
E
EF G
C H
F G
H
Reductioncenter C, scale factor 1
4
hsm11gmse_0905_t08144.ai
R T
TR
4
8
X X
hsm11gmse_0905_t08145.ai
2 2 4K
LM
JJ
K
LM
x
y
3
O
ny
y
y
x
nxxO
OP n OP
P (nx, ny)
P(x, y)
hsm11gmse_0905_t08148.ai
Got It?
enlargement. Use the ratio of the lengths of corresponding sides to �nd the scale factor.
Why is the scale
factor not 412, 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.
GEOM12_SE_CCS_C09L06.indd 588 7/5/11 2:36:46 PM
2 Common Core
HSM12GESE_CC16.indd 2 8/1/11 1:26:56 PM
1.75 in.1.75 in.
Problem 2
Got It?
Problem 3
Got It?
Lesson 9-6 Dilations 589
Finding a Dilation Image
What are the coordinates of the vertices of D2( PZG)? Graph the image of PZG.
dilation 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 . P Z G .
2. a. D12
( PZG)?
b. Reasoning How are PZ and P Z related? How are PG and P G , and GZ
and G Ze�ects of dilations on lines.
reductions, 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. �e 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.
3.
the height of the new image?
geom12_se_ccs_c09l06_t03.ai
2y
6 4 2 4OxZ
PG
2y
6 4 4Ox
Z
P
G
Z
P
G
geom12_se_ccs_c09l06_t04.ai
What are the coordinates of the vertices of D
dilation is the origin and the scale factor is 2, so use the dilation rule
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).
Biologythat is 7enlargement is 7. �e photo shows an apple seed under this magnifying glass. What is the actual length of the apple seed?
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.
GEOM12_SE_CCS_C09L06.indd 589 7/5/11 2:36:51 PM
CC-16 Dilations 3
HSM12GESE_CC16.indd 3 8/1/11 1:26:57 PM
Lesson Check
590 Chapter 9 Transformations
Practice and Problem-Solving Exercises
�e blue �gure is a dilation image of the black �gure. �e labeled point is the center of dilation. Tell whether the dilation is an enlargement or a reduction. �en �nd the scale factor of the dilation.
7. 8. 9.
10. 11. 12.
13. 14. 15.
PracticeA See Problem 1.
A 64
hsm11gmse_0905_t06790.ai
C
3
9
hsm11gmse_0905_t06791.ai
2
4
R
hsm11gmse_0905_t06792.ai
K
hsm11gmse_0905_t06793.ai
L
hsm11gmse_0905_t06794.ai
M
hsm11gmse_0905_t06795.ai
O
x
y
2
2
4
6
4 6
hsm11gmse_0905_t06796.ai
Ox
y
1
2 3
hsm11gmse_0905_t06797.ai hsm11gmse_0905_t06798.ai
x
y
O46 2
4
6
Do you know HOW?1.
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. D12 (0, 6) 4. D10 (0, 0)
Do you UNDERSTAND?5. Vocabulary
6. Error Analysis �gure is a dilation image
dilation with center A.
A. B. hsm11gmse_0905_t09439.ai
12
C8
hsm11gmse_0905_t09440.ai
2
6
1
3
A
hsm11gmse_0905_t09441.aihsm11gmse_0905_t09442.ai
MATHEMATICAL PRACTICES
MATHEMATICAL PRACTICES
GEOM12_SE_CCS_C09L06.indd 590 7/5/11 2:36:57 PM
4 Common Core
HSM12GESE_CC16.indd 4 8/1/11 1:26:58 PM
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.
20.1.36 cm.
21.
22.1.68 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, 3
2 ; D 6 (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 de�nition of scale factor of a dilation
35. Think About a Plan LMN and its image L M N for a dilation with center Px and y
What is the relationship between LMN and L M N ?What is the scale factor of the dilation?
36. Writing
See Problem 2.
hsm11gmse_0905_t06799.ai
x
y
O
2
4
P R
Q
2
51
hsm11gmse_0905_t06800.ai
x
y
O
2
1
P
Q
R
4
3
hsm11gmse_0905_t06801.ai
x
y
O
1
2P
Q
R3
See Problem 3.
ApplyB
hsm11gmse_0905_t06802.ai
x
y
O
2
4
2
W
T
Q
R13
hsm11gmse_0905_t06804.ai
L
L
M
NNPM
2
4
y(2y 60)
x 3
x
GEOM12_SE_CCS_C09L06.indd 591 7/5/11 2:37:02 PM
CC-16 Dilations 5
HSM12GESE_CC16.indd 5 8/1/11 1:26:59 PM
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
40. Copy Reduction
A dilation maps HIJ onto H I J . Find the missing values.
41. HI 8 in. H I 16 in. 42. HI 7 cm H I 5.25 cm 43. HI ft H I 8 ft
IJ 5 in. I J in. IJ 7 cm I J cm IJ 30 ft I J ft
HJ 6 in. H J in. HJ cm H J 9 cm HJ 24 ft H J 6 ft
Copy TBA and point O for each of Exercises 44–47. Draw the dilation image T B A .
44. D(2, O) ( TBA) 45. D(3, B) ( TBA)
46. D(13
, T) ( TBA) 47. D(12
, O) ( TBA)
48. Reasoning AB and its dilation image A B with A, B, A , and B
Reasoning Write true or false for Exercises 49–52. Explain your answers.
49.
50. A dilation with a scale factor greater than 1 is a reduction.
51.
52.
Coordinate Geometry In the coordinate plane, you can extend dilations to include scale factors that are negative numbers. For Exercises 53 and 54, use
PQR with vertices P(1, 2), Q(3, 4), and R(4, 1).
53. Graph D 3 ( PQR).
54. a. Graph D 1 ( PQR). b. re�ection through a point.
hsm11gmse_0905_t09542.ai
hsm11gmse_0905_t06805.ai
B
T
A
O
ChallengeC
GEOM12_SE_CCS_C09L06.indd 592 7/5/11 2:37:07 PM
6 Common Core
HSM12GESE_CC16.indd 6 8/1/11 1:27:00 PM
Lesson 9-6 Dilations 593
55. Shadows of rectangle ABCD on a wall so that each
ABCDA B C D
length of AB A B is ABCD
is the light?
hsm11gmse_0905_t06807.ai
B
C
D
AA D
BC
Mixed Review60. JKL J(23, 2), K(4, 1), and L(1, 23). What are the coordinates of
J , K , and L if (Rx a is T 2, 3 )( JKL) J K L ?
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 63. c
See Lesson 9-5.
See Lesson 7-2.
geom12_se_ccs_c09l06_t0001.ai
R
ab
T S
c
U
M
120
150
503
4.5
6
5
N Y
Z
Standardized Test Prep
56. 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
57.
A quadrilateral is not a rectangle.
A quadrilateral has no diagonals.
58.
equilateral equiangular scalene
59. Use the �gure at the right to answer the questions below.a. b.
SAT/ACT
hsm11gmse_0905_t09443.ai
ShortResponse
GEOM12_SE_CCS_C09L06.indd 593 7/5/11 2:37:12 PM
CC-16 Dilations 7
HSM12GESE_CC16.indd 7 8/1/11 1:27:01 PM
