G.SRT.A.1 STUDENT NOTES & PRACTICE WS #2 – geometrycommoncore.com
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Dilations
A dilation is a transformation that produces an image that is the same shape as the pre-image
but is a different size, either larger or smaller.
A dilation used to create an image larger than the original is an expansion or enlargement. A
dilation used to create an image smaller than the original is a contraction or reduction. A figure
and its dilation are similar figures.
You are probably familiar with the word "dilate" as it relates to the
eye. "The pupils of the eye were dilated." As light hits the eye, the
pupil enlarges or contracts depending upon the amount of light.
This concept of enlarging and contracting is "dilating".
DILATION – CONTRACTION (REDUCTION)
PRE-IMAGE
IMAGE
DILATION – EXPANSION (ENLARGEMENT)
PRE-IMAGE
IMAGE
When the dilation is an enlargement, the scale factor is greater than 1.
Scale factor of 3: sometimes written as 3:1 (image: pre-image),
Image is three times bigger proportionally to the pre-image.
When the dilation is a reduction, the scale factor is between 0 and 1.
Scale factor of ½: sometimes written as 1:2 (image: pre-image),
Image is half the size proportionally to the pre-image.
So what happens if the scale factor is 1? Nothing!!
To multiply anything by 1 maintains the pre-image. It is like rotating a shape 360°
(the shape is not altered in any way) it is an identity transformation.
G.SRT.A.1 STUDENT NOTES & PRACTICE WS #2 – geometrycommoncore.com
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So what actually happens when a shape is dilated?
The length of each side of the image is equal to the length of the corresponding side of the
pre-image multiplied by the scale factor,
A’B’ = k  AB, B’C’ = k  BC and A’C’ = k  AC. (image: pre-image)
This dilation has a scale factor of 3:1 or 3.
k=3
A’B’:AB
12: 4
3:1
B'
12 cm
4 cm
B’C’:BC
16.5: 5.5
A
3:1
A'
B
16.5 cm
5.5 cm
5 cm
15 cm
C
A’C’: AC
15:5
3:1
C'
1. Determine whether the following situations are REDUCTIONS OR ENLARGEMENTS.
a)
b)
c)
Scale Factor of 5:2
(image : pre-image)
DO,5 ( A)  A '
B
B'
C'
O
C
A'
A
Reduction or Enlargement
Reduction or Enlargement
d)
Reduction or Enlargement
e)
f)
A'
Scale Factor of
DO,0.5 ( H )  H '
A
2
3
O
Reduction or Enlargement
Reduction or Enlargement
Reduction or Enlargement
2. Determine the missing point.
D
a)
d)
M
R
T
Y
W
P
DT ,3 (Y )  (______)
b)
D
DH ,3 ( K )  (______)
e)
DT ,2 (M )  (______)
R,
1
2
( F )  (______)
K
F
H
c)
DP,4 (______)  ( R)
f)
D
F,
1
8
(______)  ( K )
G.SRT.A.1 STUDENT NOTES & PRACTICE WS #2 – geometrycommoncore.com
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The distance from the center of the dilation to each point of the image is equal to the distance
from the center of the dilation to each corresponding point of the pre-image figure times the
scale factor, OB’ = k  OB and OC’ = k  OC.
Scale Factor of k:1
Scale Factor 1:k
OB = y
OB’ = ky
ky:y
k:1
OC = x
OC’ = kx
kx:x
k:1
ky
B'
B
y
C
O
C'
x
kx
3. The dilation centered at O with a scale factor of 3. OA = 6 cm, OB = 7 cm and AB = 8 cm
a) A’B’ = _____________
B'
b) OB’ = _____________
c) OA’ = _____________
B
7
8
d) AA’ = _____________
O
6
e) BB’ = _____________
A
A'
f) What is the ratio of OA:AA’? _____________
4. T the dilation centered at O with a scale factor of
1
. OA = 6 cm, OB = 8 cm and AB = 4 cm.
2
a) A’B’ = _____________
8
b) OB’ = _____________
c) OA’ = _____________
d) AA’ = _____________
e) BB’ = _____________
f) What is the ratio of OA’:AA’? ______________
B
B'
O
6
4
A'
A
G.SRT.A.1 STUDENT NOTES & PRACTICE WS #2 – geometrycommoncore.com
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A dilation is a transformation of the plane, ,
such that if O is a fixed point, k is a non-zero real number,
and P' is the image of point P,
then O, P and P' are collinear and
.
1.
2.
P' is the image of P under a
dilation about O of ratio 2
OP' = 2OP and
ΔA'B’C’ is the image of ΔABC
under a dilation about O of ratio
OA' = OA and
OB' = OB and
OC' = OC and
G.SRT.A.1 STUDENT NOTES & PRACTICE WS #2 – geometrycommoncore.com
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1. Circle whether the following situations are REDUCTIONS OR ENLARGEMENTS.
a) Scale Factor of 7:1
b)
(image : pre-image)
Reduction
or
Enlargement
DO,1.75 ( A)  A '
d)
Reduction
or
DO,3 ( H )  H '
Reduction
or
c)
Enlargement
Reduction
e) Scale Factor of 2:3
f)
(image : pre-image)
Enlargement
g)
Reduction
or
Enlargement
D
O,
(G)  G '
Enlargement
C'
B
O
D'
D'
B'
D
B'
D
B'
B
O
B
Enlargement
J)
Reduction
or
Enlargement
Reduction
k)
F
D'
C
F'
D
E'
D
O
O
B
B'
B'
Enlargement
Enlargement
C'
E
D'
C
or
l)
C'
or
or
i)
O
Reduction
Enlargement
C
C'
or
5
3
or
Reduction
h)
C
Reduction
B'
B
O
Reduction
or
Enlargement
Reduction
O
B
or
Enlargement
2. Determine the missing point.
D
G
H
B
O
F
C
E
a)
DO,3 ( B)  (______)
b)
DO,2 ( H )  (______)
c)
DG,2 ( H )  (______)
d)
DE ,3 (C )  (______)
e)
DH ,4 (______)  ( F )
f)
DH ,9 (______)  ( E)
g)
DH ,3 (____)  (C )
h)
DC ,2.5 ( F )  (____)
i)
D
G,
7
5
( F )  (_____)
G.SRT.A.1 STUDENT NOTES & PRACTICE WS #2 – geometrycommoncore.com
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3. Determine the ratio. (Reduce the ratio)
5
A
3
C
1
D
4
E
B
a) CD : DE
_______ : _______
b) EB : BD
_______ : _______
c) CD : DA
_______ : _______
d) AC : CD
_______ : _______
e) CE : CD
_______ : _______
f) AC : AB
_______ : _______
4. Answer the following questions about the dilation, centered at O.
a) Is this an enlargement or a reduction?
__________________
Explain how you determined your answer.
A'
A
b) What scale factor do you think this is?
____________
Explain how you determined your answer.
O
B
B'
c) What angle is the same size as OBA?
Explain how you determined your answer.
____________
5. Answer the following questions about the dilation centered at O with a scale factor of 3.
OA = 3, OB = 5 and AB = 4
a) A’B’ = _____________
A'
b) OB’ = _____________
A
c) OA’ = _____________
d) AA’ = _____________ (be careful)
3
O
4
5
B
e) BB’ = _____________ (be careful)
B'
f) What is the ratio of OA:AA’? ______________