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Sec Math II

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Similar triangles are triangles that have the same shape but not necessarily the same size. ... right triangles can also be similar but use the criteria. – PowerPoint PPT presentation

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Title: Sec Math II


1
Similarity of Triangles
  • Sec Math II
  • Unit 8 Lesson 3
  • Class Notes

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enter key
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2
In geometry, two polygons are similar when one is
a replica (scale model) of the other.
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3
Consider Dr. Evil and Mini Me from Mike Meyers
hit movie Austin Powers. Mini Me is supposed to
be an exact replica of Dr. Evil.
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4
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5
The following are similar figures.
I
II
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6
The following are non-similar figures.
I
II
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7
Feefee the mother cat, lost her daughters, would
you please help her to find her daughters. Her
daughters have the similar footprint with their
mother.
Feefees footprint
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8
1.
Which of the following is similar to the above
triangle?
B
A
C
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9
Similar triangles are triangles that have the
same shape but not necessarily the same size.
?ABC ? ?DEF
When we say that triangles are similar there are
several repercussions that come from it.
?A ? ?D
?B ? ?E
?C ? ?F
10
Six of those statements are true as a result of
the similarity of the two triangles. However, if
we need to prove that a pair of triangles are
similar how many of those statements do we need?
Because we are working with triangles and the
measure of the angles and sides are dependent on
each other. We do not need all six. There are
three special combinations that we can use to
prove similarity of triangles.
1. PPP Similarity Theorem ? 3 pairs of
proportional sides
2. PAP Similarity Theorem ? 2 pairs of
proportional sides and congruent angles between
them
3. AA Similarity Theorem ? 2 pairs of
congruent angles
11
1. PPP Similarity Theorem ? 3 pairs of
proportional sides
?ABC ? ?DFE
12
2. PAP Similarity Theorem ? 2 pairs of
proportional sides and congruent angles between
them
m?H m?K
?GHI ? ?LKJ
13
The PAP Similarity Theorem does not work
unless the congruent angles fall between the
proportional sides. For example, if we have the
situation that is shown in the diagram below, we
cannot state that the triangles are similar. We
do not have the information that we need.
Angles I and J do not fall in between sides GH
and HI and sides LK and KJ respectively.
14
3. AA Similarity Theorem ? 2 pairs of
congruent angles
m?N m?R
?MNO ? ?QRP
m?O m?P
15
It is possible for two triangles to be similar
when they have 2 pairs of angles given but only
one of those given pairs are congruent.
m?T m?X
m?S m?Z
m?S 180?- (34? 87?)
?TSU ? ?XZY
m?S 180?- 121?
m?S 59?
16
Note One triangle is a scale model of the other
triangle.
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17
How do we know if two triangles are similar or
proportional?
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18
Triangles are similar () if corresponding angles
are equal and the ratios of the lengths of
corresponding sides are equal.
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19
Interior Angles of Triangles
The sum of the measure of the angles of a
triangle is 1800. Ð A Ð B ÐC 1800
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20
Example 6-1b
Determine whether the pair of triangles is
similar. Justify your answer.
Answer Since the corresponding angles have
equal measures, the triangles are similar.
21
If the product of the extremes equals the product
of the means then a proportion exists.
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22
     
This tells us that ? ABC and ? XYZ are similar
and proportional.
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23
Q Can these triangles be similar?
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24
AnswerYes, right triangles can also be similar
but use the criteria.
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25
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26
Do we have equality?
This tells us our triangles are not similar. You
cant have two different scaling factors!
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27
If we are given that two triangles are similar or
proportional what can we determine about the
triangles?
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28
The two triangles below are known to be similar,
determine the missing value X.
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29
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30
In the figure, the two triangles are similar.
What are c and d ?
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31
In the figure, the two triangles are similar.
What are c and d ?
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32
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33
Two triangles are called similar if their
corresponding angles have the same measure.
?
?
?
?
?
?
34
Two triangles are called similar if their
corresponding angles have the same measure.
?
Ratios of corresponding sides are equal.
C
A
?
a
c
?
?
?
?
b
B
a
A
b
B
c
C


35
Mary is 5 ft 6 inches tall. She casts a 2 foot
shadow. The tree casts a 7 foot shadow. How tall
is the tree?
36
Mary is 5 ft 6 inches tall. She casts a 2 foot
shadow. The tree casts a 7 foot shadow. How tall
is the tree?
Marys height
Trees height
Marys shadow
Trees shadow

x
5.5
2
7
37
Mary is 5 ft 6 inches tall. She casts a 2 foot
shadow. The tree casts a 7 foot shadow. How tall
is the tree?
5.5
x
2
7

Marys height
Trees height
Marys shadow
Trees shadow

x
5.5
2
7
38
5.5
x
2
7

7 ( 5.5 ) 2 x 38.5 2 x
x 19.25
The height of the tree is 19.25 feet
39
Congruent Figures
  • In order to be congruent, two figures must be the
    same size and same shape.

40
Similar Figures
  • Similar figures must be the same shape, but their
    sizes may be different.

41
Similar Figures
  • This is the symbol that means similar.
  • These figures are the same shape but different
    sizes.

42
SIZES
  • Although the size of the two shapes can be
    different, the sizes of the two shapes must
    differ by a factor.

4
2
6
6
3
3
2
1
43
SIZES
  • In this case, the factor is x 2.

4
2
6
6
3
3
2
1
44
SIZES
  • Or you can think of the factor as 2.

4
2
6
6
3
3
2
1
45
Enlargements
  • When you have a photograph enlarged, you make a
    similar photograph.

X 3
46
Reductions
  • A photograph can also be shrunk to produce a
    slide.

4
47
Determine the length of the unknown side.
15
12
?
4
3
9
48
These triangles differ by a factor of 3.
15 3 5
15
12
?
4
3
9
49
Determine the length of the unknown side.
?
2
24
4
50
These dodecagons differ by a factor of 6.
?
2 x 6 12
2
24
4
51
Sometimes the factor between 2 figures is not
obvious and some calculations are necessary.
15
12
10
8
18
12
?
52
To find this missing factor, divide 18 by 12.
15
12
10
8
18
12
?
53
18 divided by 12 1.5
54
The value of the missing factor is 1.5.
15
12
10
8
18
12
1.5
55
When changing the size of a figure, will the
angles of the figure also change?
?
40
?
?
70
70
56
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.
40
40
70
70
70
70
57
We can verify this fact by placing the smaller
triangle inside the larger triangle.
40
40
70
70
70
70
58
The 40 degree angles are congruent.
40
70
70
70
70
59
The 70 degree angles are congruent.
40
40
70
70
70
70
70
60
The other 70 degree angles are congruent.

4
40
70
70
70
70
70
61
Find the length of the missing side.
50
?
30
6
40
8
62
This looks messy. Lets translate the two
triangles.
50
?
30
6
40
8
63
Now things are easier to see.
50
30
?
6
40
8
64
The common factor between these triangles is 5.
50
30
?
6
40
8
65
So the length of the missing side is?
66
Thats right! Its ten!
50
30
10
6
40
8
67
Similarity is used to answer real life questions.
  • Suppose that you wanted to find the height of
    this tree.

68
Unfortunately all that you have is a tape
measure, and you are too short to reach the top
of the tree.
69
You can measure the length of the trees shadow.
10 feet
70
Then, measure the length of your shadow.
10 feet
2 feet
71
If you know how tall you are, then you can
determine how tall the tree is.
6 ft
10 feet
2 feet
72
The tree must be 30 ft tall. Boy, thats a tall
tree!
6 ft
10 feet
2 feet
73
Similar figures work just like equivalent
fractions.
30
5
11
66
74
These numerators and denominators differ by a
factor of 6.
30
5
6
11
6
66
75
Two equivalent fractions are called a proportion.
30
5
11
66
76
Similar Figures
  • So, similar figures are two figures that are the
    same shape and whose sides are proportional.

77
Practice Time!
78
1) Determine the missing side of the triangle.
?
9
5
?
3
4
12
79
1) Determine the missing side of the triangle.
15
9
5
?
3
4
12
80
2) Determine the missing side of the triangle.
36
36
6
6
?
4
?
81
2) Determine the missing side of the triangle.
36
36
6
6
?
4
24
82
3) Determine the missing sides of the triangle.
39
?
?
33
?
8
24
83
3) Determine the missing sides of the triangle.
39
13
?
33
11
8
24
84
4) Determine the height of the lighthouse.
?
?
8
2.5
10
85
4) Determine the height of the lighthouse.
?
32
8
2.5
10
86
5) Determine the height of the car.
?
?
3
5
12
87
5) Determine the height of the car.
7.2
?
3
5
12
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