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February 13, 2018
Lesson 11.1 Dilations
Key concepts:
Scale Factor
Center of Dilation
Similarity
A
B C
PreImage:
A dilation changes the size of a figure.
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A
B C
Image:
A'
B' C'PreImage:
What does a dilation NOT change?
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SHAPE!
A dilation must make all sides of a shape larger (or smaller) by the exact same amount.
This amount is called the SCALE FACTOR.A
B C
A'
B' C'
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4 5
8
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What is the scale factor that transforms the preimage in blue to the image in red?
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A
B C
A'
B' C'
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10 12
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What is the scale factor that transforms the preimage in blue to the image in red?
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A'
B' C'
A
B C
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10 12
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What is the scale factor that transforms the preimage in red to the image in blue?
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Scale Factor can be determined by the following formula:
Scale Factor =Corresponding Side of Image
Corresponding Side of PreImage
For a transformation to be a dilation, the scale factor must be the same for each corresponding part of the preimage and image.
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Determine which of the following transformations represents a dilation and which does not. Determine Scale Factor where appropriate.
PreImage
Image
2014
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710
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3
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Scale Factor can also be thought of as the constant being used to multiply the x and y coordinates when using coordinate notation for a transformation.
Example Dilating by a scale factor of 2 is the same as:
(x,y) (2x, 2y)
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(x,y) (3x, 2y)
(x,y) (1.5x, 1.5y)
(x,y) (1/2x, 1/2y)
(x,y) (3x, 1/3y)
Which of the following represent a dilation?
(x,y) (3x, 2y)
(x,y) (1.5x, 1.5y)
(x,y) (1/2x, 1/2y)
(x,y) (3x, 1/3y)
Which of the following represent a dilation?
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Dilation on a coordinate plane, example:
(1 , 4)
(2 , 1)
(4 , 3)
0 1 2 3 4 5 6 7 8 9 10
1
2
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x
y
A
B
C
Transform preimage ABC to image A'B'C' using the following coordinate notation:
(x,y) (2x, 2y)
A(1,4).................A'( )
B(2,1).................B'( )
C(4,3).................C'( )
Center of DilationWhen a shape is dilated, the center of dilation is the location from which the vector of dilation begins.
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A
B C
Image: A'
B' C'
PreImage:
Center of dilation(sometimes called the origin of dilation)
Image: PreImage:
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Think of the center of dilation as the location where you might shine a flashlight from. The preimage is then what you are shining your flashlight on, and your image is the shadow of the preimage on a screen a certain distance away.
Image: PreImage:
Center of dilation
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Here is the dilation from before find the center of dilation.
(1 , 4)
(2 , 1)
(4 , 3)
0 1 2 3 4 5 6 7 8 9 10
1
2
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x
y Transform preimage ABC to image A'B'C' using the following coordinate notation:
(x,y) (2x, 2y)
A(1,4).................A'(2,8)
B(2,1).................B'(4,2)
C(4,3).................C'(8,6)
Here is the dilation from before find the center of dilation.
(1 , 4)
(2 , 1)
(4 , 3)
0 1 2 3 4 5 6 7 8 9 10
1
2
3
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x
y
The CENTER of DILATION is at the origin: (0,0)
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Similarity, lesson 11.4Two figures are similar if all of their corresponding sides are in proportion, and their angle measures are all the same.
In other words, similar figures are the same shape but not necessarily the same size.
Dilations create similar figures.
The sides of similar figures must be in proportion; they must have the same:
SCALE FACTOR!
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Similarity, Theorem 1
Check for SSS similarity by seeing if each pair of corresponding sides of two triangles have the same scale factor.
Practice with the following shapes:
1612
11
68
5.5
3 6
8
4 8
12
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Practice with the following shapes:
1612
11
68
5.5
3 6
8
4 8
12
Not SimilarSimilar by SSS
Similarity, Theorem 2
Check for angles labeled congruent, as well as vertical angles, and alternate interior angles.
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Practice AA similarity:
Practice AA similarity:
Vertical Angles
Not SimilarSimilar by AA
Alternate Interior Angles
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Practice AA similarity:
Vertical Angles
Not SimilarSimilar by AA
Alternate Interior Angles
Similarity Theorem 3
Check for scale factor as well as one pair of congruent angles.
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Practice SAS Similarity:
Practice SAS Similarity:
Vertical Angles
Not Similar Similar by SAS
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Today you will need:
Your notes
Your "textbook"
This week:
Friday Module 11/12 test (Dilations, Similarity, and Proportions)
Thursday quiz. An A on the quiz will give you a 5% bonus on your test.
Study Guides are by the door
Module 10 retake packets are by the door.
Must be COMPLETE for a retake.
Retakes by Friday, February 23
Module 12 Proportional Relationships
Triangle Proportionality Theorem
Using Proportional Relationships to find missing lengths
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Triangle Proportionality Theorem
Page 634
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12
5
7
x
Do pages 657 and 658 all problems
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Today you will need your notes, and your study guide.
Quiz will be tomorrow (Wednesday)
Test Friday
11.2 Ratios and Directed Line Segments
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Directed Line segment
Like a vector, a directed line segment has a beginning, a direction, and an end.
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10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
Directed line segment AB
A
B
10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
Partitioning a directed line segment means to use a ratio to go a certain distance from the beginning of the line segment to the end.A
B
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10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
Example:
Partition AB by a ratio of 3:1 really means to go 3/4 of the way from A to B.
A
B
10 8 6 4 2 0 2 4 6 8 10
10987654321
12345678910
x
y
We do this by creating our right triangles, and going 3/4 of the way in the x direction and the y direction.
A
B
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Grab the homework and let's practice.
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