7-6 Similarity Transformations

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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.
Answer:
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.
Answer:
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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