Transcript Chapter8 - Catawba County Schools

Similarity
Chapter 8
8.1 Ratio and Proportion
 A Ratio is a comparison of two numbers.
o Written in 3 ways
o A to B
o A/B
o A:B
 A Proportion is an equation where two or
more ratios are equal.
a c e
o
 
b d f
Properties
 Cross Product
 If a/b = c/d then ad = bc
 Reciprocal Property
 If a/b = c/d then b/a = d/c
Geometric Mean
 The geometric mean
of two positive
numbers, a and b, is
the positive number
x, such that:
a x
 , then, x  a  b
x b
 The geometric mean
of 8 and 18 is 12
because:
8 12

12 18
and because:
(8  18)  144  12
Solve
4
10

2z  6 7 z  2
4 5

x 7
3
2

y2 y
Simplify the Ratios
12cm
4m
12cm
4cm
6 ft
8in
8.2 Problem Solving with
Proportions
 Additional Properties
 If a / b = c / d, then a / c = b / d
 If a / b = c / d, then (a + b) / b = (c + d) / d
Mini-Me and Dr. Evil
Mini Horse and Pony
Cheetah Mother with
Babies
Find the width to length
ratio on each figure.
16mm
10cm
20mm
7.5cm
Find the missing lengths
24
6
20
16
9
3
8.3 Similar Polygons
 When all corresponding angles are
congruent and lengths of corresponding
sides are proportional, the two polygons
are similar.
 The symbol ~ is used to indicate
similarity.
Scale Factor
 If two polygons are similar, then the ratio
of the lengths of two corresponding sides
is called the scale factor.
x
5
16
3.5
Theorem
 If two polygons are
similar, then the ratio
of their perimeters is
equal to the ratios of
their corresponding
side lengths.
L
K
M
Q
P
O
R
N
T
S
KL  LM  MN  NO  OK KL LM MN NO OK





PQ  QR  RS  ST  TP
PQ QR
RS
ST
TP
Similarity
 Are ABCD and EFGH similar?
A
B
E
H
3.5
D
7
C
4
G
 What is the scale factor?
2
F
8.4 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.
Similarity
 PQR ~ _____
 PQ = QR = RP
L
P
 20 =
18
.
12
 y = ____
 x = ____
12
x
y
Q
M
20
15
R
N
Similarity
 Are the two triangles similar?
57
92
92
41
Similarity
 Are the two triangles similar?
65
65
50
8.5 Proving Triangles are
similar
 Side-Side-Side (SSS) Similarity Theorem
 If the lengths of the corresponding sides of two
triangles are proportional, then the triangles are
similar.
P
A
IF:
AB BC CA


PQ QR RP
THEN: ABC ~ PQR
R
Q
B
C
Side-Angle-Side
 (SAS) Similarity Theorem
 If an angle of one triangle is congruent to an
angle of a second triangle and the lengths of the
sides including these angles are proportional,
then the triangles are similar.
THEN:
IF:
X  M
and
ZX
XY

PM MN
XYZ ~ MNP
X
M
Z
Y
P
N
Examples
 Pg 492 #1-5
8.6 Proportions and
similar triangles
 Four Proportionality Theorems.
Triangle Proportionality
Theorem
 If a line parallel to one side of a triangle
intersects the other two sides, then it divides
the two sides proportionally.
Q
IF:
TU  QS
T
R
RT RU
THEN:

TQ US
S
U
Converse of the Triangle
Proportionality Theorem
 If a line divides two sides of a triangle
proportionally, then it is parallel to the
third side.
Q
IF:
THEN:
RT RU

TQ US
T
R
TU  QS
S
U
Theorems
 If three parallel lines intersect two
transversals, then they divide the
transversals proportionally.
 If r ll s and s ll t and l and m intersect r, s,
UW VW

and t, then
.
r
WY
s
XZ
t
U
W
Y
V
X
Z
l
m
Theorems
 If a ray bisects an angle of a triangle, then it
divides the opposite side into segments
whose lengths are proportional to the
lengths of the other two sides.
A
If
CD
then
bisects
ABC
AD CA

DB CB
D
C
B
Examples
 Pg 502 #1-5