Download Geo Unit 7

Geometry
Unit 7
Proportions and Similarity
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Unit
Ratios and Proportions
Ratio
Proportion
a to b
a is to b as c is to d
a:b
a : b :: c : d
a
b
a c

b d
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Properties of Proportions
Cross-Product Property
If
a c

b d
, then
ad  bc
Reciprocal Property
If
a c

b d
, then
b d

a c
Congruent Proportions
a c

b d
a b

c d
d c

b a
b d

a c
a b c d

b
d
ab cd

b
d
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Similar Polygons
•
A  D, B  E, C  F
•
AC AD BC


DF DE EF
F
C
A
B
D
E
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Similar Triangles
• Angle-Angle (AA) Similarity Postulate
– if two angles of one triangle are congruent to
two angles of another triangle, then the two
triangles are similar
E
B
A
C
D
F
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Similar Triangles
• Side-Side-Side (SSS)
– if the corresponding sides of two triangles are
proportional, then the triangles are similar
E
B
A
C
D
F
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Similar Triangles
• Side-Angle-Side (SAS)
– if an angle of one triangle is congruent to an
angle of a second triangle, and the sides
including these angles are proportional, then
the triangles are similar
E
B
A
C
D
F
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Transversal Proportionality
• Transversal Proportionality Theorem
– Parallel lines divide transversals proportionally.
l m n
a

b
a
AC
b
AC
c
d
c

BD
d

BD
A
a
B
l
c
m
b
C
d
D
n
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Similar Triangles
• Given
ABC
DEF
– A  D, B  E, C  F
AC AD BC


–
DF DE EF
F
C
A
B
D
E
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Fractals
• A geometric figure in which a pattern is
repeated so that certain parts of the figure are
similar to each other.
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Special Segments of
Similar Triangles
• Altitudes
– the lengths of corresponding altitudes are
proportional to the lengths of the
corresponding sides
ABC
FGH , then
A
AD AB

FJ FG
F
C
D
B H
J
G
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Special Segments of
Similar Triangles
• Angle bisectors
– the lengths of corresponding angle bisectors
are proportional to the lengths of the
corresponding sides
KLM
QRS , then
R
LP LM

RT RS
L
Q
T
S K
P
M
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Special Segments of
Similar Triangles
• Medians
– the lengths of corresponding medians are
proportional to the lengths of the
corresponding sides
ABC
WXY , then
C
CD AB

YZ WX
Y
A
D
B W
Z
X
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Triangle Angle Bisector Theorem
• an angle bisector in a triangle separates
the opposite side into two segments that
are proportional to the lengths of the other
two sides
R
ST SR

QT QR
Q
T
S
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Ratios of Figures
• Ratio of Perimeters: equal to ratio of sides
– a:b
• Ratio of Areas: equal to ratio of sides squared
– a2 : b2
• Ratio of Volumes: equal to ratio of sides cubed
– a3 : b3
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