Download Lesson 8.3 Similar

Sec Math II
Unit 8 Lesson 3
Class Notes
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In geometry, two polygons
are similar when one is a
replica (scale model) of
the other.
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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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The following are similar figures.
I
II
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The following are non-similar figures.
I
II
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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.
Feefee’s
footprint
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1.
Which of the following is similar to the
above triangle?
A
B
C
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Similar triangles are triangles that have the same
shape but not necessarily the same size.
A
D
E
B
F
C
ABC  DEF
When we say that triangles are similar there are several
repercussions that come from it.
A  D
B  E
C  F
AB
DE
=
BC
EF
=
AC
DF
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
E
1. PPP Similarity Theorem
 3 pairs of proportional sides
A
9.6
5
B
C
12
mAB

mDF
mBC

mFE
5
 1 .25
4
12
 1.25
9 .6
F
4
mAC
13

 1.25
mDE 10.4
ABC  DFE
D
2. PAP Similarity Theorem
 2 pairs of proportional sides and congruent
angles between them
L
G
70
H
7
I
mGH
5

 0 .66
7 .5
mLK
mHI
7

 0 .66
mKJ 10.5
70
J
10.5
mH = mK
GHI  LKJ
K
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.
L
G
50
H
7
I
J
50
K
10.5
Angles I and J do not fall in between sides GH and HI and
sides LK and KJ respectively.
3. AA Similarity Theorem
 2 pairs of congruent angles
Q
M
70
50
N
mN = mR
mO = mP
O
50
70
P
MNO  QRP
R
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.
T
X
Y
34
34
59
59
Z
87 59
U
S
mS = 180- (34 + 87)
mS = 180- 121
mS = 59
mT = mX
mS = mZ
TSU  XZY
Note: One triangle is a scale model of the
other triangle.
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How do we know if two
triangles are similar or
proportional?
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Triangles are similar (~)
if corresponding angles
are equal and the ratios
of the lengths of
corresponding sides are
equal.
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Interior Angles of Triangles
B
A
C
The sum of the measure
of the angles of a
triangle is 1800.
C 1800
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Determine whether the pair of triangles is similar.
Justify your answer.
Answer: Since the corresponding angles have equal
measures, the triangles are similar.
If the product of the
extremes equals the
product of the means
then a proportion
exists.
a c

b d
bc  ad
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This
 ABC and AC
 XYZ are
ABtells us thatBC
=K
=K
=K
similar and proportional.
YZ
XZ
XY
12
2
6
8
2
4
10
2
5
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Q: Can these triangles be similar?
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Answer—Yes, right triangles can also
be similar but use the criteria.
AB
=
XY
AC
BC
=K
=
XZ
YZ
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AB
=
XY
BC
=
YZ
AC
=K
XZ
6
8
10
=
=
=K
4
6
8
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Do we have equality?
6
8
10
=
=
=K
4
6
8
6
8
= 1.5 but
= 1.3
4
6
This tells us our triangles are not
similar. You can’t have two different
scaling factors!
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If we are given that two
triangles are similar or
proportional what can we
determine about the
triangles?
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The two triangles below are known to
be similar, determine the missing value
X.
7.5 4.5

5
x
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7.5 4.5

5
x
54.5  7.5x
22.5  7.5x
3 x
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In the figure, the two triangles are similar.
What are c and d ?
A
5
P
10
c
B
R
4
d
Q
6
10 c

5 4
C
40  5c
8c
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In the figure, the two triangles are similar.
What are c and d ?
A
5
P
10
c
B
R
4
d
Q
6
C
10 6

30  10d
5 d
3d
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Sometimes we need to measure a distance
indirectly. A common method of indirect
measurement is the use of similar triangles.
17 6

102 h
36  h
h
6
17
102
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Two triangles are called “similar”
if their corresponding angles have
the same measure.






Two triangles are called “similar”
if their corresponding angles have
the same measure.
Ratios of corresponding
sides are equal.
C

a
A
A
a
c





b
=
B
b
B
=
c
C
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?
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?
Mary’s height
Tree’s height
=
Mary’s shadow
Tree’s shadow
x
5.5
2
7
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?
Mary’s height
Tree’s height
=
5.5
2
=
x
7
Mary’s shadow
Tree’s shadow
x
5.5
2
7
5.5
=
x
7
7 ( 5.5 ) = 2 x
38.5
= 2x
x
= 19.25
The height of the tree is 19.25 feet
2
Congruent Figures
• In order to be congruent, two
figures must be the same size
and same shape.

Similar Figures
• Similar figures must be the
same shape, but their sizes may
be different.

Similar Figures
This is the symbol that
means “similar.”
These figures are the same
shape but different sizes.


SIZES
• Although the size of the two
shapes can be different, the
sizes of the two shapes4must
differ by a factor.
2
3
1
3

6
6
2
SIZES
• In this case, the factor is x 2.
4
2
3
3
1

6
6
2
SIZES
• Or you can think of the factor
as
2.
4
2
3
3
1

6
6
2
Enlargements
• When you have a photograph
enlarged, you make a similar
photograph.

X3
Reductions
• A photograph can also be
shrunk to produce a slide.
4

Determine the length of the
unknown side.
15
12

?
4
3
9
These triangles differ by a factor
of 3.
15
3= 5
15
12

?
4
3
9
Determine the length of the
unknown side.
?
2
4

24
These dodecagons differ by a
factor of 6.
?
2
4

24
Sometimes the factor between 2
figures is not obvious and some
calculations are necessary.
15
12

18
?=
8
10
12
To find this missing factor,
divide 18 by 12.
15
12

18
?=
8
10
12
18 divided by 12
= 1.5
The value of the missing
factor is 1.5.
15
12

18
1.5 =
8
10
12
When changing the size of a
figure, will the angles of the
figure also change?
?
40
70
70
?
?
Nope! Remember, the sum of all 3
angles in a triangle MUST add to 180
degrees.
If the size of the
angles were
40
increased,
40
the sum
would exceed
180
degrees.
70
70
70
70
We can verify this fact by placing
the smaller triangle inside the
larger triangle.
40
40
70
70
70
70
The 40 degree angles
are congruent.
40
70
70
70
70
The 70 degree angles
are congruent.
40
40
7070
70
70
70
The other 70 degree
angles are congruent.
4
40
70 7070
7070
Find the length of the missing
side.
50
?
30
6
40
8
This looks messy. Let’s
translate the two triangles.
50
?
30
6
40
8
Now “things” are easier to see.
50
30
?
6
40
8
The common factor between
these triangles
is 5.
50
30
?
6
40
8
So the length of
the missing side
is…?
That’s right! It’s ten!
50
30
10
6
40
8
Similarity is used to answer real
life questions.
• Suppose that you
wanted to find the
height of this tree.
Unfortunately all that
you have is a tape
measure, and you are
too short to reach the
top of the tree.
You can measure the length of
the tree’s shadow.
10 feet
Then, measure the length of your
shadow.
10 feet
2 feet
If you know how tall you are,
then you can determine how tall
the tree is.
6 ft
10 feet
2 feet
The tree must be 30 ft tall. Boy,
that’s a tall tree!
6 ft
10 feet
2 feet
Similar figures “work” just like
equivalent fractions.
30
66
5
11
These numerators and
denominators differ by a factor of
6.
30
66
6
6
5
11
Two equivalent fractions are
called a proportion.
30
66
5
11
Similar Figures
• So, similar figures are
two figures that are the
same shape and whose
sides are proportional.
Practice Time!
1) Determine the missing side of
the triangle.
?
5
9
3
4
12
1) Determine the missing side of
the triangle.
15
5
9
3
4
12
2) Determine the missing side of
the triangle.
6
6
36
36
4
?
2) Determine the missing side of
the triangle.
6
6
36
36
4
24
3) Determine the missing sides of
the triangle.
39
?
33
?
8
24
3) Determine the missing sides of
the triangle.
39
13
33
11
8
24
4) Determine the height of the
lighthouse.
?
8
2.5
10
4) Determine the height of the
lighthouse.
32
8
2.5
10
5) Determine the height of the
car.
?
3
5
12
5) Determine the height of the
car.
7.2
3
5
12