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Module11and12 Smartnotes.notebook 1 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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Module11and12 Smartnotes.notebook

1

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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February 13, 2018

A

B C

Image:

A'

B' C'Pre­Image:

What does a dilation NOT change?

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February 13, 2018

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'

3

4 5

8

6

10

What is the scale factor that transforms the pre­image in blue to the image in red?

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February 13, 2018

A

B C

A'

B' C'

6

10 12

9

18

What is the scale factor that transforms the pre­image in blue to the image in red?

15

A'

B' C'

A

B C

6

10 12

12

24

What is the scale factor that transforms the pre­image in red to the image in blue?

20

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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

18

710

9

3

6

3

7

9

18

21

9

10 11

14

15 16

21

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

3

4

5

6

7

8

9

10

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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February 13, 2018

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

3

4

5

6

7

8

9

10

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

4

5

6

7

8

9

10

x

y

The CENTER of DILATION is at the origin: (0,0)

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February 13, 2018

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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February 13, 2018

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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February 13, 2018

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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February 13, 2018

Practice SAS Similarity:

Practice SAS Similarity:

Vertical Angles

Not Similar Similar by SAS

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February 13, 2018

Today you will need:

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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February 13, 2018

Triangle Proportionality Theorem

Page 634

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12

5

7

x

Do pages 657 and 658 ­ all problems

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February 13, 2018

Quiz will be tomorrow (Wednesday)

Test Friday

11.2 ­ Ratios and Directed Line Segments

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February 13, 2018

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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February 13, 2018

Grab the homework and let's practice.

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February 13, 2018

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