### 7-6 Similarity Transformations

```7-6 Similarity Transformations
p. 511
You identified congruence transformations.
• Identify similarity transformations.
• Verify similarity after a similarity
transformation.
Definitions
Transformation – an operation that maps an original
figure (preimage) onto a new figure (image).
Dilation – a transformation that enlarges or reduces
the original figure proportionally.
Center of dilation – a fixed point used in a dilation
Scale factor of a dilation – the extent of the dilation.
Scale factor – the ratio of a length on the image to a
corresponding length on the preimage.
The letter “k” usually represents the scale factor of a
dilation. The value of “k” determines whether the
dilation is an enlargement or a reduction.
A dilation is a type of similarity transformation since
it produces a similar figure.
p. 511
Determine whether the dilation
from Figure A to Figure B is an
enlargement or a reduction. Then
find the scale factor of the dilation.
B is smaller than A, so the
dilation is a reduction.
The distance between the vertices at (2, 2) and
(2, –2) for A is 4 and from the vertices at (1, 1)
and (1, –1) for B is 2.
Answer: So, the scale factor is _2 or _1 .
4
2
Determine whether the
dilation from Figure A to
Figure B is an enlargement
or a reduction. Then find the
scale factor of the dilation.
B is larger than A, so the
dilation is an enlargement.
The distance between the vertices at (3, 3) and
(–3, 3) for A is 6 and from the vertices at (1, 1)
and (–1, 1) for B is 2.
Answer: So, the scale factor is _6 or 3.
2
Determine whether the dilation from Figure A to
Figure B is an enlargement or a reduction. Then
find the scale factor of the dilation.
A. reduction; _1
_2
_1
B. reduction; _3
C. enlargement; 2
D. enlargement; 3
B. Determine whether the dilation from Figure A
to Figure B is an enlargement or a reduction. Then
find the scale factor of the dilation.
A. reduction; _2
3
_1
B. reduction; 3
_3
C. enlargement; 2
D. enlargement; 2
PHOTOCOPYING A photocopy of a receipt is
1.5 inches wide and 4 inches long. By what percent should the
receipt be enlarged so that its image is
2 times the original? What will be the dimensions of the
enlarged image?
To enlarge the receipt 2 times the original, use a scale
factor of 2. Written as a percent, the scale factor is
(2 ● 100%) or 200%. Now, find the dimensions of the
enlarged receipt.
width: 1.5 in. ● 200% = 3 in.
length: 4 in. ● 200% = 8 in.
The enlarged receipt will be 3 inches by
8 inches.
PHOTOGRAPHS Mariano wants to enlarge a picture he took
that is 4 inches by 7.5 inches. He wants it to fit perfectly
into a frame that is 400% of the original size. What will be
the dimensions of the enlarged photo?
A. 15 inches by 25 inches
B. 8 inches by 15 inches
C. 12 inches by 22.5 inches
D. 16 inches by 30 inches
Verifying Similarity after a Dilation
If you want to verify that a dilation produces a
similar figure, you can compare corresponding
angles and sides.
For triangles, use SAS Similarity
A. Graph the original figure and its dilated image. Then verify
that the dilation is a similarity transformation.
original: M(–6, –3), N(6, –3), O(–6, 6)
image: D(–2, –1), F(2, –1), G(–2, 2)
Graph each figure. Since M and D are
both right angles, M  D. Show that
the lengths of the sides that include M
and D are proportional.
Use the coordinate grid to find the lengths of the vertical
segments MO and DG and the horizontal segments MN and DF.
Answer: Since the lengths of the sides that include
M and D are proportional,
ΔMNO ~ ΔDFG by SAS Similarity.
B. Graph the original figure and its
dilated image. Then verify that the
dilation is a similarity
transformation.
original: G(2, 1), H(4, 1), I(2, 0), J(4, 0)
image: Q(4, 2), R(8, 2), S(4, 0), T(8, 0)
Since the figures are rectangles, their
corresponding angles are congruent.
Find and compare the ratios of corresponding sides.
A. Graph the original figure and its dilated image.
Then determine the scale factor of the dilation.
original: B(–7, –2), A(5, –2), D(–7, 7)
image: J(–3, 0), K(1, 0), L(–3, 3)
1
A. __
2
1
B. __
3
3
C. __
2
3
D. __
4
B. Graph the original figure and its dilated image.
Then determine the scale factor of the dilation.
original: A(4, 3), B(6, 3), C(4, 2), D(6, 2)
image: E(6, 4), F(10, 4), G(6, 2), H(10, 2)
A. 2
1
B. __
3
C. 3
D. 4
7-6 Assignment
Page 514, 6-13, 15
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