Clusters: 1. Understand similarity in terms of similarity transformations 2. Prove theorems involving similarity. 3. Define trigonometric ratios and solve problems involving right triangles. 4. Apply trigonometry to general triangles.

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Clusters 1. Understand similarity in terms of
similarity transformations 2. Prove theorems
involving similarity. 3. Define trigonometric
ratios and solve problems involving right
triangles. 4. Apply trigonometry to general
triangles.
Similarity, Right Triangles, and Trigonometry
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Learning Target

1. I can define dilation.
2. I can perform a dilation with a given center and
   scale factor on a figure in the coordinate plane.

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Connection to previews lesson

• Previously, we studied rigid transformations, in
  which the image and preimage of a figure are
  congruent. In this lesson, you will study a type
  of nonrigid transformation called a dilation, in
  which the image and preimage of a figure are
  similar.

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Dilations

• A dilation is a type of transformation that
  enlarges or reduces a figure but the shape stays
  the same.
• The dilation is described by a scale factor and a
  center of dilation.

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Dilations

• The scale factor k is the ratio of the length of
  any side in the image to the length of its
  corresponding side in the preimage. It describes
  how much the figure is enlarged or reduced.

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The dilation is a reduction if k lt 1 and it is an
enlargement if k gt 1.
P
P
6
5
P
P
2
3
Q
Q
R


C
C
R
Q
R
Q
R
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Constructing a Dilation

• Examples of constructed a dilation of a triangle.

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Steps in constructing a dilation

• Step 1 Construct ?ABC on a coordinate plane
  with A(3, 6), B(7, 6), and C(7, 3).

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Steps in constructing a dilation

• Step 2 Draw rays from the origin O through A,
  B, and C. O is the center of dilation.

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Steps in constructing a dilation

• Step 3 With your compass, measure the distance
  OA. In other words, put the point of the compass
  on O and your pencil on A. Transfer this
  distance twice along OA so that you find point A
  such that OA 3(OA). That is, put your point
  on A and make a mark on OA. Finally, put your
  point on the new mark and make one last mark on
  OA. This is A.

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Steps in constructing a dilation

• Step 3

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Steps in constructing a dilation

• Step 4 Repeat Step 3 with points B and C.
• That is, use your compass to find points B
  and C such that OB 3(OB) and OC 3(OC).

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Steps in constructing a dilation

• Step 4

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Steps in constructing a dilation

• You have now located three points, A, B, and
  C, that are each 3 times as far from point O as
  the original three points of the triangle.
• Step 5 Draw triangle ABC.
• ?ABC is the image of ABC under a dilation
  with center O and a scale factor of 3. Are these
  images similar?

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Steps in constructing a dilation

• Step 5

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Questions/ Observations

• Step 6 What are the lengths of AB and AB? BC
  and BC? What is the scale factor?

• AB 4
• AB 12
• BC 3
• BC 9

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Questions/ Observations

• Step 7 Measure the coordinates of A, B, and C.

Image A(9, 18) B(21, 18) C(21, 9)
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Questions/ Observations

• Step 8 How do they compare to the original
  coordinates?

P(x, y) ? P(kx, ky)
Pre-image A(3, 6) ? B(7, 6) ? C(7, 3)
?
Image A(9, 18) B(21, 18) C(21, 9)
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In a coordinate plane, dilations whose centers
are the origin have the property that the image
of P(x, y) is P(kx, ky).
SOLUTION
Because the center of the dilation is the origin,
you can find the image of each vertex by
multiplying its coordinates by the scale factor.
C
D
A
D
B
A(2, 2) ? A(1, 1)
B(6, 2) ? B(3, 1)
B

C(6, 4) ? C (3, 2)
D(2, 4) ? D(1, 2)
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In a coordinate plane, dilations whose centers
are the origin have the property that the image
of P(x, y) is P(kx, ky).
SOLUTION
From the graph, you can see that the preimage has
a perimeter of 12 and the image has a perimeter
of 6.
C
D
A
D
B
A preimage and its image after a dilation are
similar figures.
B

Therefore, the ratio of the perimeters of a
preimage and its image is equal to the scale
factor of the dilation.
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Example 3

• Determine if ABCD and ABCD are similar
  figures. If so, identify the scale factor of the
  dilation that maps ABCD onto ABCD as well as
  the center of dilation.

Is this a reduction or an enlargement?
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Assignment/Homework

• Work with a partner in the classwork on
  Constructing Dilation
• Homework
• Answer Guided Practice page 510 12 to 15.