Name __________________________
CC Geometry H
HW #7
Date: _____________________
1. Use the side splitter method to divide SM into 4 equal sized pieces.
M
S
2. Use ray X and the dilation method to divide DM into 6 equal sized pieces.
D
X
3. Dilate hexagon D3
D2
D1
D4
D5
M
from center using a scale factor of = / .
H
I
I'
H'
O
D'
G
G'
D
E'
E
F'
F
over
4. If the segment below represents the interval from zero to one on the number
line, locate and label 2 .
5
2/5
0
1
5. Using your compass, draw the dilation of parallelogram ABCD from center O
using scale factor n = 2.
D'
D
O
C
C'
A
B
A'
B'
` 6. Given circle C with radius CR and circle C', find the center of dilation O.
R'
R
C'
C
O
Aim #8: How do dilations map figures from two different
CC Geometry H
centers with the same scale factor ?
Do Now: Dilate A(2,-3) using scale factor 2 and center (0,0). Label the image A'.
1
Dilate B(1,4) using scale factor /2 and center (-7,-2) Label the image B'.
Dilate C(8,-2) using scale factor 3 and center (10,-4). Label the image C'.
Activity: Dilating a segment using two different centers of dilation
Graph and label AB with endpoints A(1,3) and B(3,-2).
a) Using the origin (0,0) as the center of dilation, dilate AB using scale factor 2.
Connect A' and B', and label the segment A'B'.
b) Using the point (-3,2) as the center of dilation, dilate AB using scale factor 2.
Connect A" and B" and label the segment A"B".
c) What appears to be true about A'B' and A"B"?
d) Justify your conjecture from (c), by determining the slopes and lengths of
A'B' and A"B".
Slope of A'B' = _____ Slope of A"B" = ______ A'B' = ____ A"B" = ____
e) Name the translation that maps A'B' onto A"B".
_____________
1
The figure below was dilated using centers O1 and O2,, each using scale factor /2.
• Locate centers O1 and O2.
• What seems to be true about the two dilated figures compared to each other?
• What transformation maps drawing #2 onto #3?
• How is the translation vector related to the line joining the two centers?
Drawing 1
Drawing 2
Drawing 3
The following conclusions are true when a figure is dilated using the same scale
factor and two different centers of dilation:
(1) The dilation of a figure from two different centers by the same scale factor
yields congruent figures.
(2) The dilated figures are congruent by a translation along a vector that is
parallel to the line through the centers.
Exercise (1): Triangle ABC has been dilated with scale factor 2 from centers O1
and O2. What can you say about line segments A1A2, B1B2, and C1C2?
B2
A2
B1
A1
C2
B
A
O1
C1
C
O2
You will now create your own dilation of a figure using two different centers.
Exercise (2): Using a compass or ruler, dilate AB by a factor of 2.
First use O1 as the center, and label the dilated segment A1B1.
Then use O2 as the center, and label the dilated segment A2B2.
A
O2
O1
B
What do you notice about the translation vector that maps the scale drawings to
each other relative to O1O2 , the segment that passes through the centers?
Composition of Dilations: A first figure is dilated to a second figure; the
second figure is dilated to a third figure.
What is the relationship between the 1st and 2nd scale factors compared to the
3rd scale factor in a composition of three dilations?
Exercise (3): Drawing 2 is a scale drawing of Drawing 1 with scale factor r1.
Drawing 3 is a scale drawing of Drawing 2 with scale factor r2
.
Drawing 1
B3
Drawing 3
A3
A1
B2
A2
C3
Drawing 2
C1
C2
Determine the relationship between Drawing 3 and Drawing 1:
a) Locate O1 , the center of dilation of Drawing 1 to Drawing 2
Locate O2, the center of dilation of Drawing 2 to Drawing 3.
Locate O3, the center of dilation of Drawing 1 to Drawing 3.
Observation: The three centers of dilation are ____________________.
b) Estimate the scale factor for:
,the dilation of Drawing 1 to Drawing 2. _______
, the dilation of Drawing 2 to Drawing 3. _______
, the dilation of Drawing 1 to Drawing 3. _______
c) What is the relationship between the first two scale factors and the third
scale factor? ________________________________________________
In a series of dilations, the scale factor that maps the final image is the
product of all the scale factors in the series of dilations.
B1
2
Exercise (4): ΔABC has been dilated with scale factor /3 from center O1 to get
1
ΔA'B'C'. ΔA'B'C' is dilated from center O2 with scale factor /2 to get ΔA"B"C".
Find where the center of dilation is that maps ΔABC to ΔA"B"C". On what
segment will it lie? __________ What is the scale factor for this dilation?
B'C'D'.
2 and
Exercise (5): Figure W is dilated from O1 with a scale factor r1 = 2 to yield W'.
Figure W' is then dilated from center O2 with a scale factor r2 = 1/4 to yield W".
If you were to dilate figure W, what scale factor would be required to yield an
image that is congruent to figure W"?
Let's Sum it Up!
A composition of two dilations,
and
, is a dilation with scale factor
and center on O1O2. There is a translation parallel to the line passing through the
centers of the individual dilations.
Name: ____________________
Date: _____________
CC Geometry H
HW #8
1. Using a compass or ruler, dilate CD by a factor of 2
First use O1 as the center, and label the dilated segment C1D1.
Then use O2 as the center, and label the dilated segment C2D2.
D
O2
C
O1
What do you notice about the translation vector that maps the scale drawings to
each other relative to O1O2 , the line that passes through the centers?
1
2. A dilation with center O1 and scale factor /2 maps figure F to figure F'. A
3
dilation with center O2 and scale factor /2 maps figure F' to F". Draw figures F'
and F", and then find the center O3 and scale factor r of the dilation from F to F".
F
O1
O2
OVER
3) Determine the image points which represent the dilations given:
r
Mixed Review:
5. ≮4 ≅ ≮5 and SE bisects ≮RSF.
Find m≮1.
E
4
F
1
T
2 3
5
R
S
E
6. ΔDEF is the image of ΔABC after a
B
dilation with center O. If AD = 30, AC = 9,
O
DF = 15, find OA.
C
A
F
D
7. Below, A'B' is the image of AB under a dilation from point O with an unknown
scale factor. Use your ruler to determine the scale factor r.
A' A
B'
B