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Unit 7 Lesson 8.7: Dilations
CC STANDARDS
SRT 1: Verify experimentally the properties of dilations given by a center and a scale factor.
GMD 5: Know that the effect of a scale factor k greater than zero on length is to multiply by k; determine length using scale factors.
LESSON GOALS
• Define and identify a dilation as an
enlargement or a reduction.
• Use properties of dilations to solve for
variables.• Draw a dilation given
the scale factor. ESLRs: Becoming Effective Communicators, Competent Learners and Complex Thinkers
definition
IsometryA transformation that preserves lengths. Also known as a “rigid” transformation. (Figures remain congruent)
Reflections, rotations and translations are all isometries.
definition
Scale FactorThe ratio of the lengths of two corresponding sides for similar polygons
48
3032
20
48
32
3
2
30
20
3
2
Scale factor
key vocabulary
II
definition
Dilationa “nonrigid” transformation that maps a preimage onto an image that is similar.
P
P
definition
DilationReduction: the scale factor comparing the image to the preimage is between 0 and 1
I
0 1k
is the scale factork
P
I
I
P
k
I
definition
DilationEnlargement: the scale factor comparing the image to the preimage is greater than 1
P1k
is the scale factork
PI
I
P
k
example
Determine whether the dilation from Figure A to Figure B is a reduction or an enlargement. Then, find the values of the variables.
AB
5
5
3 3
12.5
y
xz
I
definition
Describing a Dilation scale factor center of dilation
a fixed point in the plane about which all points are expanded or contracted.
P
In an enlargement, the preimage is between the center and the image.
I
definition
Describing a Dilation scale factor center of dilation
a fixed point in the plane about which all points are expanded or contracted.
P
In a reduction, the image is between the center and the preimage.
I
definitionDescribing a Dilation
Scale factor can be found by comparing corresponding side lengths Or by comparing the distance from the center to corresponding points Always compare the image to the preimage
P
PI
I
P
k
PdId
I
P
dk
d
example
A
BD
A’
B’D’
Find the scale factor. Tell whether the dilation is a reduction or an enlargement. 21.5
in
47 in21.5
47k
215
470
43
94
C
example
Find the scale factor. Tell whether the dilation is a reduction or an enlargement. Then find the values of the variables.
example
53
Center: 0,0
k x
y
53
3,3 ' 5,5A A
A’
53
3,0 ' 5,0B B
B’
53
0, 6 ' 0, 10C C
A
B
C
C’
Draw a dilation of the figure using the given scale factor.
example
12
Center: 0,0
k
x
y
12
1 12 21,5 ' , 2D D
D’
12
4,4 ' 2,2E E
E’
12
125,2 ' 2 ,1F F
F’
12
122,3 ' 1,1G G
DE
FG
G’
1
1
2
2
summaryWhat determines if a dilation is an enlargement or
a reduction?
How do you find the scale factor of a dilation?
A friend asks you with help on the following problem; Draw the dilation of the figure using the given scale factor.
Explain to your friend how they should solve this problem.
Today’s Assignment
worksheet