7-2: Exploring Dilations and Similar Polygons

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7-2 Exploring Dilations and Similar Polygons
Expectation G3.2.1 Know the definition of
dilation and find the image of a figure under a
dilation. G3.2.2 Given two figures that are
images of each other under some dilation,
identify the center and magnitude of the dilation.
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Size Changes

• When we start with one figure and make it bigger
  or smaller, it is called a size change
  transformation.
• The original figure is called the preimage and
  the resulting figure is called the image.

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• The magnitude, k, of a size change is how many
  times bigger (or smaller) the image is than the
  preimage.

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Types of Dilations

• Contraction reduction the image is smaller than
  the preimage magnitude is greater than 0, but
  less than 1.
• Expansion enlargement the image is larger than
  preimage magnitude is greater than 1.

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Dilations
The following terms all indicate a size
change dilation dilitation contraction expansion
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• A picture is enlarged by a scale factor of 125
  and then enlarged again by the same scale factor.
  If the original picture was 4 x 6, how large is
  the final copy?

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• By what scale factor was the original picture
  enlarged?

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Size Changes With Coordinates

• To perform a size change of given magnitude on a
  polygon with known coordinates, multiply the
  magnitude of the size change by each of the
  coordinates of the polygon.

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• If D(x,y) (3x,3y), what is the image of the
  point (-5,8)?
• What is the scale factor of the dilation?

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• A triangle has coordinates A(3,-1), B(4,3) and
  C(2,5). The triangle will undergo a dilation
  using a scale factor of 3. Determine the
  coordinates of the vertices of the resulting
  triangle.

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• Triangle ABC is a dilation of triangle XYZ. Use
  the coordinates of the 2 triangles to determine
  the scale factor of the dilation.
• A(-1, 1), B(-1, 0), C(3,1)
• X(-3, 3), Y(-3, 0), Z(9, 3)

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• Triangle XYZ is a dilation of ?ABC. Use the
  coordinates of the 2 triangles to determine the
  scale factor of the dilation.
• A(-1, 1), B(-1, 0), C(3,1)
• X(-3, 3), Y(-3, 0), Z(9, 3)

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Size Change Distance Theorem

• The image of a segment transformed by a dilation
  with scale factor k is parallel to and k times
  the length of the preimage.

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• Before a size change, the slope of AB is 4 and AB
  8. After a size change of magnitude .5, what is
  the slope of AB and AB?

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Center of a Dilation

• The center of any dilation is where the lines
  through all corresponding points intersect.

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C is the center of the dilation mapping ?XYZ onto
?LMN
C
Y
X
Z
M
L
N
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• Given two figures which are dilations of each
  other, how can you find the center of the
  dilation?

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Determine the center of the dilation.
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Similar Figures
Defn Two figures, F and G, are similar (written
F G) iff corresponding angles are congruent and
corresponding sides are proportional. Dilations
always result in similar figures!!!
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Similar Figures
If WXYZ ABCD, then
?W ? ?A ?X ? ?B ?Y ? ?C ? Z ? ?D
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• If ?ABC is similar to ?DEF in the diagram below,
  then m?D ?
• 80
• 60
• 40
• 30
• 10

B
E
80
D
F
40
A
C
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Determine whether the triangles are similar.
Justify your response!
9.75
13
9
12
3.75
5
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Scale Factor
The scale factor (magnitude) between similar
figures is the ratio of the lengths of
corresponding sides.
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Triangle ABC is similar to triangle DEF.
Determine the scale factor of DEF to ABC (be
careful the order is important), then calculate
the lengths of the unknown sides.
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In the figure below, ?ABC is similar to ?DEF.
What is the length of DE? A. 12 B. 11 C. 10 D.
7? E. 6?
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Assignment

• pages 351-353,
• 13 25 (odds), 29 and 41