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Lesson 6-4 Similar Figures and Proportions
供稿人: Warren
because the ratios of their matching sides
made a proportion.
In the last example, the triangles were similar
What is the simplified form of all of these ratios?
15
10
Their ratios were all equal.
=
12
8
=
9
6
=
L
4
O
Is rectangle LMNO similar to rectangle HIJK?
10
10
N
M
4
3
H
K
8
8
I
J
3
L
4
O
Is rectangle LMNO similar to rectangle HIJK?
1. Are the corresponding (matching)
10
angles congruent (the same size)?
10
Yes.
N
M
4
No.
3
H
K
8
8
I
J
3
L
4
O
Is rectangle LMNO similar to rectangle HIJK?
1. Are the corresponding (matching)
2. Do the corresponding (matching) sides
form a proportion (make equal fractions)?
angles congruent (the same size)?
10
10
Yes.
N
M
4
3
H
K
8
8
I
J
3
L
4
O
Is rectangle LMNO similar to rectangle HIJK?
Which side corresponds to LM?
10
10
N
M
4
3
H
K
8
8
I
J
3
L
4
O
Is rectangle LMNO similar to rectangle HIJK?
Which side corresponds to MN?
10
10
N
M
4
3
H
K
8
8
I
J
3
L
4
O
Is rectangle LMNO similar to rectangle HIJK?
Write the ratio of LM to HI:
10
10
N
M
4
3
H
K
8
8
I
J
3
L
4
O
Is rectangle LMNO similar to rectangle HIJK?
Write the ratio of MN to IJ:
10
10
N
M
4
3
H
K
8
8
I
J
3
L
4
O
Is rectangle LMNO similar to rectangle HIJK?
Use cross multiplication to determine
if the ratio LM:HI is equal to the ratio MN:IJ.
10
10
10
8
?
N
M
4
4
3
3
H
K
Yes
No
8
8
I
J
3
L
4
O
Is rectangle LMNO similar to rectangle HIJK?
10 • 3 = 30
8 • 4 = 32
if the ratio LM:HI is equal to the ratio MN:IJ.
Use cross multiplication to determine
10
10
30 ≠ 32
N
M
4
3
H
K
NO!
8
8
I
J
3
L
4
O
Now answer the question:
Is rectangle LMNO similar to rectangle HIJK?
10
10
Yes
No
N
M
4
3
H
K
8
8
I
J
3
L
4
O
Now answer the question:
Is rectangle LMNO similar to rectangle HIJK?
2. The sides do not make a proportion.
1. The angles are congruent.
10
10
NO! 
N
M
4
3
H
K
8
8
I
J
3
A
8
60∘
B
Is parallelogram ABCD similar to
parallelogram KLMN?
120∘
12
12
120∘
D
60∘
8
C
K
4
60∘
L
120∘
6
120∘
6
60∘
N
4
M
A
8
60∘
1. Are the corresponding angles congruent?
B
Is parallelogram ABCD similar to
parallelogram KLMN?
120∘
12
12
120∘
Yes.
No.
D
60∘
8
C
K
4
60∘
L
120∘
6
120∘
6
60∘
N
4
M
A
8
60∘
2. Do the sides form a proportion?
Complete this ratio:
B
Is parallelogram ABCD similar to
parallelogram KLMN?
120∘
12
12
120∘
D
60∘
8
C
8
K
4
60∘
L
120∘
6
120∘
6
60∘
N
4
M
A
Complete this proportion:
8
60∘
2. Do the sides form a proportion?
B
Is parallelogram ABCD similar to
parallelogram KLMN?
120∘
12
12
120∘
D
60∘
8
C
8
4
K
4
60∘
=
L
120∘
6
120∘
6
60∘
N
4
M
A
Are the cross-products
equal?
8
60∘
2. Do the sides form a proportion?
B
Is parallelogram ABCD similar to
parallelogram KLMN?
120∘
Yes.
No.
12
12
120∘
D
60∘
8
C
8
4
K
4
60∘
=
L
120∘
6
12
6
120∘
6
60∘
N
4
M
What is the cross-product?
A
8
60∘
2. Do the sides form a proportion?
B
Is parallelogram ABCD similar to
parallelogram KLMN?
120∘
12
12
120∘
D
60∘
8
C
8
4
K
4
60∘
=
L
120∘
6
12
6
120∘
6
60∘
N
4
M
A
8
60∘
B
Is parallelogram ABCD similar to
parallelogram KLMN?
120∘
12
12
Yes.
No.
120∘
D
60∘
8
C
K
4
60∘
L
120∘
6
120∘
6
60∘
N
4
M
Which side corresponds to YZ?
CD
AB
AD
XW
Complete the proportion:
YZ
2
=
equivalent fractions to solve for YZ.
Use cross multiplication or
YZ
2
=
16
24
YZ = 
24
D
A
12
C
B
15
Given that
trapezoid ABCD ~ trapezoid WZYX,
find m.
W
m
Z
16
Y
X
24
D
A
12
C
B
15
Complete the proportion used to find m:
Given that
trapezoid ABCD ~ trapezoid WZYX,
find m.
W
m
16
24
Z
16
=
Y
X
24
D
A
12
C
B
15
Given that
trapezoid ABCD ~ trapezoid WZYX,
find m.
W
m
Use cross multiplication or 
equivalent fractions to find m.
16
24
Z
=
16
m
12
m = 
Y
X
24
D
A
12
C
B
15
trapezoid ABCD ~ trapezoid WZYX,
find YZ.
Given that
W
Complete the ratio for YZ:
m
Z
16
Y
X
YZ
24
D
A
12
C
B
15
trapezoid ABCD ~ trapezoid WZYX,
find YZ.
Given that
W
Complete the proportion:
m
Z
16
Y
X
YZ
15
=
24
D
A
12
C
B
15
trapezoid ABCD ~ trapezoid WZYX,
find YZ.
Given that
Use cross-multiplication to find YZ.
W
15
YZ
m
=
Z
16
24
16
Y
YZ = 
X
C
Given ∆ABC ~ ∆DEF
Find n.
12
10
A
B
11
F
6
n
D
E
5.5
Complete this proportion:
C
Given ∆ABC ~ ∆DEF
Find n.
12
10
A
B
11
10
n
F
6
n
=
D
E
5.5
Solve this proportion:
C
n = 
Given ∆ABC ~ ∆DEF
Find n.
12
10
A
B
11
10
n
F
6
n
=
D
E
5.5
12
6
How long is side JL?
How long is side JK?
15
GOOD JOB!
10
15
How long is side KG?
Complete the proportion:
15
t
=
t = 
Solve the proportion for t:
15
t
=
10
6
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