Exploring Dilations I
1. In the figure below ∆𝐴𝐵𝐶 has been dilated to obtain ∆𝐴′𝐵′𝐶′. ∆𝐴𝐵𝐶 has been enlarged by a scale factor
of 2 with the center of dilation at O (the origin).
12
B'
10
8
6
B
4
A'
2
20
15
10
O
5
C'
C
A
5
10
15
20
2
4
6
8
We will use the dilation shown above to discover some properties of dilations.
a. In the table below, list the coordinates of the points:
A:
A’:
B:
B’:
C:
C’:
What do you notice about the coordinates of corresponding points for a dilation centered at the
origin?
b. Find the lengths of the following segments:
̅̅̅̅:
𝐴𝐵
̅̅̅̅̅̅:
𝐴′𝐵′
̅̅̅̅ :
𝐵𝐶
̅̅̅̅̅̅:
𝐵′𝐶′
̅̅̅̅
𝐴𝐶 :
̅̅̅̅̅
𝐴′𝐶′:
What do you notice about the lengths of the corresponding segments?
c. What else do you notice about the corresponding segments of the triangles?
d. Draw a line through each set of corresponding points and the center of dilation (i.e. Draw a line
through A and A’, extending the line to the center of dilation. Draw a different line through B and
B’, extending the line to the center of dilation. Do this for all corresponding points.
What can you say about lines that pass through corresponding points of a figure that has been
dilated?
e. Find the following distances:
̅̅̅̅:
𝑂𝐵
̅̅̅̅̅:
𝑂𝐵′
̅̅̅̅
𝑂𝐴:
̅̅̅̅̅
𝑂𝐴′:
̅̅̅̅ :
𝑂𝐶
̅̅̅̅̅:
𝑂𝐶′
What do you notice?
2. In the figure below, ABCD was dilated to obtain A’B’C’D’. The dilation has a center at the origin and
1
scale factor 3.
10
B
C
C'
A
D
8
6
4
B'
2
A'
20
15
10
O
5
D'
5
10
15
2
4
6
8
10
Verify that the properties discovered in the previous example hold true for this dilation.
20
3. Find the center of dilation and scale factor for the following dilation. Explain how you found your
answers.
10
8
6
4
2
J
J'
K'
20
15
10
5
K
5
10
15
20
M'
M
2
L'
4
L
6
8
10
4. Find the center of dilation and scale factor for the following dilation. Explain how you found your
answers.
10
8
6
4
2
T'
20
15
R'=T
R
10
5
5
2
S
S'
4
6
8
10
10
15
20
5. Dilate ∆𝐷𝐸𝐹 by a scale factor of 3 and center of dilation at the origin.
10
8
6
4
E
2
20
15
10
5
5
D
10
15
20
10
15
20
F
2
4
6
8
10
Verify in multiple ways that your image meets the criteria of the dilation.
6. Dilate RSTU by a scale factor of 2 and center of dilation at the origin.
10
8
6
4
U
T
2
R
20
15
10
5
5
2
S
4
6
8
10
Verify in multiple ways that your image meets the criteria of the dilation.
1
7. Dilate ABCD by a scale factor of 2 and center of dilation at the origin.
10
8
C
6
B
4
2
20
15
10
5
5
2
10
15
20
D
4
A
6
8
10
Verify in multiple ways that your image meets the criteria of the dilation.
8. Find the center of dilation and scale factor for the following dilation. Explain how you found your
answers.
10
8
B'
6
B
4
2
C'
20
15
10
C
A'=A
5
5
2
4
6
8
10
10
15
20
Exploring Dilations II
1
1. Dilate WXYZ by a scale factor of 2 and center of dilation at (3, -2).
10
8
W
6
X
4
2
20
15
10
Z
5
Y
5
10
15
20
10
15
20
2
4
6
8
10
2. Dilate RSTU by a scale factor of 2 and center of dilation at (-3, 5).
10
8
6
4
U
T
2
R
20
15
10
5
5
2
4
6
8
10
S
3. Consider how you could use a dilation to
create a similar object to the one shown at
the right that’s twice the size and located
entirely in Quadrant II.
a. Describe in words your strategy:
b. Implement your strategy on the
graph.
c. Compare your results to those of other students. What is the same? What is different? What
accounts for the differences?
Exploring Dilations III