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Notes7.1: Solving Ratios and Proportions
Ratio: a comparison of two quantities.
3
, 3 to 4, 3:4
4
Proportion: two ratios that are equal to
each other.
3 6

4 8
Ex: A scale model of a car is 4 in. long. The actual car is 15 ft
long. What is the ratio of the length of the model to the length
of the car?
Ex: Two cities are 3
1
in. apart on a map with the scale 1 in. =
2
50 mi. Find the actual distance.
A. Solving Proportions
 Fraction bust.
 Be sure to use FOIL when you are multiplying binomials together
Solve for x:
2 3

x 12
1.)
2.)
x3 4

2
3
3.)
x 1 x  5

2x  3
2x
B. Properties of Proportions
Complete:
4.)
If
x 3
 , then 7 x  ____
2 7
6.)
If a : 2  5 : 3, then 3a  ____
8.)
If
x y
x
 , then  ____
5 2
y
10.) For the given figure, it is given that:
KR = 6, KT = 10, KS = 8
T
If
5 3
 , then 3x  ____
x 2
7.)
If
x 2
y
 , then  ____
y 9
x
9.)
If
x y
x3
 , then
 ____
3 4
3
KR KS

. Solve for the missing lengths.
KT KU
RT = ____
K
R
5.)
SU = ____
KU = ____
S
U
Notes 7.2: Similar Polygons
Similar polygons have the same _________ but not necessarily the same _____________________.
Example of similar triangles:

B
Their corresponding angles are
___________________
10
6

Y
3
A
C
8
X
5
4
Their corresponding sides are
_______________________
Z

This ratio is called a _______________
and in this case is _______
We show that they are similar with this statement: ___________________

1.)
ABCDE
A' B ' C ' D ' E '
b) mA '  _____ , mD  _____
a) scale factor = _______
9
C
x 160
mC '  _____
D
B
6
c) x = _____, y = ______, z = ______
100
A
E
8y
C'
y
D'
30
2
B'
4
A'
E'
z
The figures are similar. Solve for the variables. (Hint: redraw the diagram as two figures)
2.)
3.)
18
12
12
10
y
x
x
z
8
16
y
9
16
24
Algebra Review: Simplify.
4.)
3  15
5.)
2 8 3 12
6.)
3 5 
7.)
3
2
8.)
4 10
2
9.)
5 2
6
2
Notes7.3: Similar Triangles
 Similar triangles have: ________________ corresponding angles and ______________sides
3 Methods for Proving Triangles Similar:

You can conclude that two triangles are similar if:
_________: two pairs of corresponding angles are congruent
_________: all three pairs of sides are in the same proportion
_________: two pairs of sides are the same proportion and their included angles are congruent
Are the triangles similar? If so, state the similarity and the postulate you used.
 Re-draw the triangles in matching positions
 Mark congruent angles
small medium l arg e


small medium l arg e

Test sides for a constant proportion:

Look for these patterns: AA~, SSS~, SAS~
1.)
ABC ~  by _________ or Not Similar
F
B
35
60
2.)
ONP ~  by _________ or Not Similar
D
O
85
E
A
N
60
P
C
M
Q
3.)
ABC ~  by _________ or Not Similar
4.)
ABC ~  by _________ or Not Similar
B
F
E
15
4
4
6
6
F
D
A
8
B
C
D
10
C
12
A
5.)
QRS ~  by _________ or Not Similar
6.)
ABC ~  by _________ or Not Similar
R
Q
B
5
4
Y
10
S
6
8
70
9
X
A
Y
7.5
70
15
C
X
10
Z
6
Z
State whether the figures are always, sometimes, or never similar:
7.) two squares
9.) two rectangles
11.) two pentagons
8.) two congruent triangles
12.) two regular octagons
10.) two rhombuses
Algebra Review: Factor completely.
13.)
14.)
4 p 4r 3s 2  16 p 2r 3s 4
3x 2  15 x  21
16.)
Algebra Review: Solve by factoring.
3x 2  5 x  2  0
17.)
18.)
2 x 2  10 x  28
2
19.) 12 x  7 x  10
2 x3  8 x 2  8 x
Notes 7.5: Proportional Lengths
A. Triangle Proportionality
 A parallel slice cuts a triangle’s sides proportionally ( Side-Splitter Theorem)
Example: Solve for x
a
c
k
j
a

b
,
a

c
a

c
a
,

j
d

k
,
12
33
d
b
20
x
j

b
B. Angle Bisector Proportionality
 An angle bisector proportionally divides the opposite side
Example: Solve for x:
z

y
x
21
x

w
8
14
w

z
C. Parallel Line Proportionality
 Parallel lines proportionally divide their transversals
Example: Solve for x:
a
c
a

b
b
d
d

c
a

c
18
9
24 - x
x
Solve for the variables:
1.)
2.)
14
24
9
6
x
18
x
4
60
3.)
4.)
20
12
x
x
18
21
x+2
5.)
25
6.)
16
3
12
4
x
12
2x
7.5
Algebra Review:
7.) Write the equation of a line
in slope intercept form
that is perpendicular to
y  2 x  5 and contains the
point ( -6, 7)
8.) Write the equation of a line
that is parallel to 3 x  6 y  5
and contains the point ( 1, 4).
9.) Write the equation of a line
that contains ( -2, -3) and
( 4, -9) in standard form.