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Chapter 6: Similarity Perform Similarity Transformations 6.7

Chapter 6: Similarity Perform Similarity Transformations 6.7

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Page 1: Chapter 6: Similarity Perform Similarity Transformations 6.7

Chapter 6: Similarity

Perform Similarity Transformations 6.7

Page 2: Chapter 6: Similarity Perform Similarity Transformations 6.7

Transformations

• Remember previously we talked about 3 types of CONGRUENCE transformations, in other words, the transformations performed created congruent figures

• Rotation, Reflection, and translation

• Now we will also discussed Dilations

Page 3: Chapter 6: Similarity Perform Similarity Transformations 6.7

a dilation is a transformation in which a figure and its image are similar

the ratio of the new image to the original figure is called the scale factor

a dilation with a scale factor greater than 1 is called an enlargement

a dilation with a scale factor less than 1 is called a reduction

a figure is enlarged or reduced with respect to a fixed point called the center of dilation.

Page 4: Chapter 6: Similarity Perform Similarity Transformations 6.7

Enlargements•When you have a photograph enlarged, you make a similar photograph.

X 3

Page 5: Chapter 6: Similarity Perform Similarity Transformations 6.7

Reductions•A photograph can also be shrunk to produce a slide.

4

Page 6: Chapter 6: Similarity Perform Similarity Transformations 6.7

Determine the length of the unknown side.

12

9

15

4

3

?

Page 7: Chapter 6: Similarity Perform Similarity Transformations 6.7

These triangles differ by a factor of 3.

12

9

15

4

3

?

15 3= 5

Page 8: Chapter 6: Similarity Perform Similarity Transformations 6.7

DILATIONS

•REDUCTION VS. ENLARGEMENT•For k, a scale factor we write (x, y) (kx, ky)•If 0 < k < 1 the dilation is a reduction•If k > 1 the dilation is an enlargement

Page 9: Chapter 6: Similarity Perform Similarity Transformations 6.7

Yesterday you constructed an enlargement similar to this

•Let ABCD have A(2, 1), B(4, 1), C(4, -1), D(1, -1)•Scale factor: 2

You used the GM to show that they were similar.

Page 10: Chapter 6: Similarity Perform Similarity Transformations 6.7

Center of Dilation(Usually the origin)

•To show that two objects are similar, we draw a line from the origin to the object further away, if the line passes through the closer object in the similar vertex then the objects are similar.

Page 11: Chapter 6: Similarity Perform Similarity Transformations 6.7

When changing the size of a figure, will the angles of the figure also change?

? ?

?

70 70

40

Page 12: Chapter 6: Similarity Perform Similarity Transformations 6.7

Nope! Remember, the sum of all 3 angles in a triangle MUST add to 180 degrees.If the size of the angles were increased,the sum would exceed 180 degrees.

70 70

40

70 70

40

Page 13: Chapter 6: Similarity Perform Similarity Transformations 6.7

70 70

40

We can verify this fact by placing the smaller triangle inside the larger triangle.

70 70

40

Page 14: Chapter 6: Similarity Perform Similarity Transformations 6.7

70 70

70 70

40

The 40 degree angles are congruent.

Page 15: Chapter 6: Similarity Perform Similarity Transformations 6.7

70 707070 70

40

40

The 70 degree angles are congruent.

Page 16: Chapter 6: Similarity Perform Similarity Transformations 6.7

70 707070 70

40

4

The other 70 degree angles are congruent.

Page 17: Chapter 6: Similarity Perform Similarity Transformations 6.7

Find the length of the missing side.

30

40

50

6

8

?

Page 18: Chapter 6: Similarity Perform Similarity Transformations 6.7

This looks messy. Let’s translate the two triangles.

30

40

50

6

8

?

Page 19: Chapter 6: Similarity Perform Similarity Transformations 6.7

Now “things” are easier to see.

30

40

50

8

?

6

Page 20: Chapter 6: Similarity Perform Similarity Transformations 6.7

The common factor between these triangles is 5.

30

40

50

8

?

6

Page 21: Chapter 6: Similarity Perform Similarity Transformations 6.7

So the length of the missing side is…?

Page 22: Chapter 6: Similarity Perform Similarity Transformations 6.7

That’s right! It’s ten!

30

40

50

8

10

6

Page 23: Chapter 6: Similarity Perform Similarity Transformations 6.7

1)  Take your shoe size (no half sizes, round up).2)  Multiply your number by 5.3)  Add 50.4)  Multiply by 20.5)  Add 1013 to that number.6)  Subtract the year you were born (ex: 1996).7)  Let me guess what you got!!

WARM-UP

R U A GEN!US?

Page 24: Chapter 6: Similarity Perform Similarity Transformations 6.7

Similarity is used to answer real life questions.

•Suppose that you wanted to find the height of this tree.

Page 25: Chapter 6: Similarity Perform Similarity Transformations 6.7

Unfortunately all that you have is a tape measure, and you are too short to reach the top of the tree.

Page 26: Chapter 6: Similarity Perform Similarity Transformations 6.7

You can measure the length of the tree’s shadow.

10 feet

Page 27: Chapter 6: Similarity Perform Similarity Transformations 6.7

Then, measure the length of your shadow.

10 feet 2 feet

Page 28: Chapter 6: Similarity Perform Similarity Transformations 6.7

If you know how tall you are, then you can determine how tall the tree is.

10 feet 2 feet6 ft

Page 29: Chapter 6: Similarity Perform Similarity Transformations 6.7

The tree must be 30 ft tall. Boy, that’s a tall tree!

10 feet 2 feet6 ft

Page 30: Chapter 6: Similarity Perform Similarity Transformations 6.7

Similar figures “work” just like equivalent fractions.

530

66 11

Page 31: Chapter 6: Similarity Perform Similarity Transformations 6.7

These numerators and denominators differ by a factor of 6.

530

66 11

6

6

Page 32: Chapter 6: Similarity Perform Similarity Transformations 6.7

Two equivalent fractions are called a proportion.

530

66 11

Page 33: Chapter 6: Similarity Perform Similarity Transformations 6.7

Similar Figures

•So, similar figures are two figures that are the same shape and whose sides are proportional.

Page 34: Chapter 6: Similarity Perform Similarity Transformations 6.7

(Page 371 on Blue Text)

Page 35: Chapter 6: Similarity Perform Similarity Transformations 6.7