Name_________________________________________________________ Date_____________ Hour______
9.1 – Using Ratios and Proportions
A _______________________ is a comparison of two quantities.
The ratio of a to b can be expressed as
a
b
or
Examples:
Write each ratio in simplest form1.
2.
4.
75
400
a to b
or
a:b
3.
5. six days to two weeks
6. 24 inches : 3 feet
7. 45 centimeters to 7 meters
8. 17 yards to 15 feet
9. 280 seconds : 6 minutes
10.
75 meters to 5 kilometers
A _______________________ is an equation that shows two equivalent fractions.
There are three methods to determine if a ratio forms a proportion.
Method 1
Simplify the fractions
Method 2
Determine the decimals
Method 3
Cross Multiply
So, the answer is “YES” since the fractions, the decimals, and the cross product are equal.
1
Examples:
Determine whether the following are a proportion:
11.
12.
In the proportion below there are two cross-products.
11 and x
_____________
16 and 44
____________
11 44
=
16
x
You can use cross-multiplication to solve equations in proportion form…
Cross-multiplying:
If
a=c
b d
, then
ad =bc
Examples:
Solve each proportion by using cross-products.
1.
4.
9
x
=
28 84
5
8

k  17 152
2.
3 4x
=
18
7
3.
5.
3
4

b  16 48
x +5 x +3
=
7
5
2
Geometry G
Ratios Worksheet 1
Name ________________________
Express each ratio in lowest terms.
1. 8 to 16 ________
4.
140
________
35
2. 12 to 4 ________
3. 15 : 75 ________
5. 150 to 15 ________
6.
48
________
40
Write each ratio in lowest terms.
7. 15 milliliters to 24 liters ________
8. 6 feet to 15 inches ________
9. 75 cm to 4 m
10. 3 days to 9 hours _________
11.
________
A soccer team played 25 games and won 17.
a. What is the ratio of the number of wins to the number of loses?
b. What is the ratio of the number of games played to the number of games won?
12.
In a senior class, there are b boys and g girls. Express the ratio of the number of boys to the
total number in the class.
13.
Two numbers are in a ratio of 5 : 3. Their sum is 80. Find the largest number.
14.
Mr. Smith and Mr. Kelly are business partners. They agreed to divide the profits in the ratio of 3 : 2.
The profit amounted to $24,000. How much did each person receive?
3
Geometry G
Ratios Worksheet 2
Name ________________________
Period ______ Date _____________
Express each ratio in lowest terms.
1.
45
________
135
4. 72 to 60 ________
2. 96 : 100 ________
5.
17
________
51
3. 625 to 125 ________
6. 49 : 35 ________
7 15 kg to 90 kg ________
8 18 feet to 4 yards ________
9. 45 meters to 80 meters ________
10. 10 seconds to 2 minutes ________
11.
The Yankees won 125 games, the Red Sox won 97 games, and the Mets won 86 games. What is the
ratio of wins of the Yankees to the Red Sox to the Mets?
12.
The measure of the angles of a triangle are in a ratio of 2 : 3 : 4. Find the number of degrees in the
smallest angle of the triangle.
Do the following pairs form a proportion?
13.
120
3
and
90
4
14.
125
25
and
35
7
15.
4
9
and
5
12
4
Geometry G
Ratios Worksheet 3
Solve each proportion.
Name ________________________
Period ______ Date _____________
Circle
your final answer.
1.
5 x

6 9
2.
2
x

8 20
3.
 8 12

11
x
4.
3
20

x  35
5.
x 3 7

4
8
6.
x 6 7

5
12
7.
8 x 2

9
6
8.
1
2

x 5 3
9.
8
4

x  10 2 x  7
5
Geometry G
Ratios Worksheet 4
Name ________________________
Period ______ Date _____________
Applications of Proportions
1. A recipe for 3 dozen cookies calls for 4 cups of
flour. How much flour is needed to make 5 dozen
cookies?
2. A certain medication calls for 250 mg for every 75
lbs of body weight. How many milligrams of
medication should a 220-lb person take?
3. A 2-inch wound requires 9 inches of suture thread.
How long of a thread should a nurse have ready to
close a 5-inch wound?
4. An apartment building has 24 identical
apartments. It took 42.7 gallons of paint to paint 3
apartments. How many gallons of paint are needed
to paint 21 apartments?
Do the following ratios form a proportion? Meaning, are they equal?
1.
2 16

3 24
2.
9 10

5 18
3.
7 21

4 14
4.
8 24

7 21
6
Sect 9.2 - Changing the Size of Figures
These figures are similar
Similar Figures ~
These are not similar
Two polygons are similar if and only if the____________________
angles are ____________________ and the measures of the _______________________ sides
are ___________________________.
The symbol __________ means similar.
ABC ~ DEF (“triangle ABC is similar to triangle DEF”)
Corresponding Angles
are _______________
Corresponding Sides
have _____________________
 ______   ______
_______  _______
 ______   ______
_______  _______
 ______   ______
_______  _______
Scale Factor If the scale factor > 1,
If the scale factor < 1,
Example: Find the dimensions of the figure ...
a) using a scale factor of 2.
7
6
6
10
6
4.5
1
.
2
8
9.5
7
b) using a scale factor of
8
10
7
Similar figures are enlargements or reductions of each other. The amount of enlargement or
reduction needed to change one figure to the other is called the _________________ . The
ratio of the lengths of the corresponding sides of similar figures is the
______________________.
Determine if the polygons are similar. Show work to justify your answer.
16
1)
2)
12
8
5
3)
7
110
4
12
70
4
70
6
110
5
16
15
24
12
9
14
12
12
9
12
12
15
Find the values of x and y if JHI~MLN.
H
a) Write proportions for the corresponding sides.
2x - 3
45
J
I
50
b) Write the proportion
to solve for x.
c) Write the proportion
to solve for y.
L
15
y
M
30
N
8
A
Example: ABCD is similar to WXYZ
The similarity ratio of ABCD to WXYZ is ________.
The scale factor of ABCD to WXYZ is _________.
12
9
D
W
B
Label the lengths of the missing sides.
4
9
12
Z
X
C
3
Y
ABCDE is similar to QRSTU
The similarity ratio of ABCDE to QRSTU is __________.
2
A
2
B
E
The scale factor of ABCDE to QRSTU is _________.
3
D 4
Find the length of each side.
RS ____________
C
Q
QU ___________
QR ___________
3
R
U
15
ST ____________
T
S
Perimeter of ABCDE______________
Perimeter of QRSTU______________
ratio of perimeter of ABCDE to perimeter of QRSTU _______________
9
10
Geometry
Chapter 11.1 Scale Factor Worksheet 1
Name
Scale factor of 3
Scale factor of 2/3
Scale factor of 3/4
11
12
Geometry
Chapter 11.1 Scale Factor Worksheet 2
Name
Goal: To be able to draw a figure with a given scale factor.
Scale factor of 2
Scale factor of
1
2
13
Scale factor: 4
Scale factor:
1
3
Scale factor:
3
2
14
Geometry
Chapter 11.1 Similar Figures Worksheet 1
Name
1. Given ABCD ~ WXYZ
a. What angles are congruent?
X
B
A
W
C
b. Write the proportions that are equal.
D
Y
Z
R
2. Given XYZ~RST
a. What angles are congruent?
X
T
S
b. Write the proportions that are equal.
Y
Z
3. Explain why the figures are similar and write the similarity statement.
A
C
B
a.
E
D
H
F
What is the scale factor
from left to right?
b. What is the scale factor
from right to left?
G
15
16
Geometry
Chapter 11.1 Similar Figures Worksheet 2
Name
Determine whether the figures are similar. If yes, what is the scale factor that transforms the figure on the
left to the figure on the right? Assume the angles are congruent.
1.
Similar ?
yes
no
If yes, scale factor (left to right) _____
2.
Similar?
yes
no
If yes, scale factor (left to right)____
50
25
70
60
60
70
14
36
36
14
20
100
25
50
3.
Similar ?
yes
no
If yes, scale factor (left to right) _____
4.
Similar?
yes
no
If yes, scale factor (left to right)____
50
40
20
42
42
48
16
48
16
36
44
50
88
37.5
37.5
36
5.
Similar ?
yes
no
If yes, scale factor (left to right) _____
36
6.
Similar?
yes
no
If yes, scale factor (left to right)____
12
36
12
70
20
60
90
16
48
36
12
35
45
10
20
30
60
17
18
Geometry
Chapter 11.2 Similar Triangles Worksheet 3
Name
Goal is to understand notation related to similarity and then apply this notation to find a missing side of similar
triangles.
Definition of Similar Polygons:
Two polygons are similar if and only if the corresponding angles are
congruent and the corresponding sides are proportional.
1.
PBS ~ FOX
These corresponding angles are congruent:
B
O
_______  ________
_______  ________
P
_______  ________
S
These corresponding sides are proportional:
F
2.
X
FAN ~ PIE
N
These corresponding angles are congruent:
_______  ________
_______  ________
F
A
E
P
3.
_______  ________
These corresponding sides are proportional:
I
CAR ~ BUS
Which angles are congruent?
What sides are proportional?
19
4.
DOT ~ BAT
What proportions are equal?
B
D
52
30
8
y
O
A
T
x
6
T
Find x
5.
Find y
ARM ~ LEG
x
A
What sides are proportional?
56.7
G
L
M
y
67.5
7.5
E
R
Find x:
6.
D
CAT ~ DOG
57
Find AC and OG.
O
A
7
2
T
94
141
G
C
20
Geometry
Chapter 11.2 Similar Triangles Worksheet 4
Name
Find the missing lengths of the similar triangles.
1. ABC ~ DEF
F
Step 1: Write the corresponding sides of ABC and DEF as a proportion:
15
E
A

13

32
D
26
B
C
Step 2: Fill in the numbers and solve for the missing side.
BC = _____________
FD = _____________
2. MAC ~ GET
A
Step 1: Write the corresponding sides of ABC and DEF as a proportion:
45


M
C
39
Step 2: Fill in the numbers and solve for the missing side.
G
15
E
11
T
AC = _____________
TG = _____________
21
3. MNP ~ SQV
Step 1: Write the corresponding sides of ABC and DEF as a proportion:
Q
45


S
63
Step 2: Fill in the numbers and solve for the missing side.
N
15
19
P
M
PM = ____________
V
QV = ____________
4. ABD ~ FEC
F
BD = x – 1
8
A
CE = x + 2
3
B
C
BD = _________
D
E
EC = _________
22
Geometry
Chapter 11.2 Similar Triangles Worksheet 5
Name
Find the missing lengths. (You may get decimals.)
2. DEF ~ RST
1. MAC ~ GET
D
A
15
14
G
M
R
E
28
T
12
C
18
AC = _________
36
F
GE = _________
24
RS = _________
S
18
E
T
TR = _________
3. RST ~ WYZ
S
6.5
4.5
Y
R
Z
15
YZ = _________
8
T
W
WY = _________
23
Similarity
Geometry G
Round Table
Names________________________
______________________________
Find the missing lengths.
1.
A
2.
MAC ~ GET
E
30
R
20
M
C
20
G
S
RST ~ WYZ
8
4
T
9
Y
T
16
W
AC = ___________
WY = ___________
EG = ___________
YZ = ___________
3.
D
4.
DEF ~ RST
N
R
MNP ~ SQV
10
13
x+3
16
P
M
S
F
Z
22.5
14
10
15
T
E
S
Q
2x + 4
V
x = ___________
RS = ___________
NP = ___________
TR = ___________
QV = ____________
24
Section 9.3 Notes
Methods of proving Triangles Similar
We will look at ways of proving triangles similar.
Recall what similarity means: 1) Corresponding angles are____________
2) The ratios of the measures of corresponding sides are_______________
Postulate: AA to prove triangles similar
Given two corresponding angles congruent, can you prove the triangles similar by AA?
W
T
N
E
R
I
Therefore, AA is a way to prove triangles similar.
The other two ways to prove triangles similar are:
Theorem:
Theorem:
Don’t forget the
~ when proving similar triangles by the three above methods!
*The 3 ways to prove similar triangles are: ________, ________, and ________.
Examples
Decide if each pair of triangles is similar. If they are, write the correspondence in the first blank and the
reason in the second blank. If they are NOT similar, write NS in the second blank.
1) ∆ ABC ~ ∆ _________ by __________
C
D
18
12
7.5
5
A
12
B
F
8
E
25
2) ∆ ABC ~ ∆ _________ by __________
T
A
48
V
G
42
C
B
3)
Y
∆ YXS ~ ∆ _________ by __________
A
12
8
15
40
X
10
S
40
Z
N
4) ∆ ABC ~ ∆ _________ by __________
D
18
A
8
C
6
B
12
E
26
Geometry
Chapter 11.2 Justifying Similar Triangles Worksheet 6
Name
Determine whether each pair of triangles is similar. If the triangles are similar, justify your answer by using
SSS, SAS, and AA. Make sure you have work to support your answer.
1.
I
T
2.5
4
3
4.8
Z
K
1
X
1.6
J
Yes
No
 ___________________ ~  ____________________ by ____________________
2.
D
F
75˚
45˚
60˚
Yes
3.
60˚
H
G
No
E
C
 ____________________ ~  ____________________ by ___________________
P
V
6
W
7
9
U
O
Yes
No
8
Q
 ____________________ ~  ____________________ by ___________________
27
B
8.
J
10
L
18
15
18
12
C
K
A
27
 ____________________ ~  ____________________ by ___________________
Yes
No
9.
Ryan is 5 feet tall. His shadow is 9 feet long and the shadow of a building is 36 feet long. How
tall is the building? Draw two similar triangles and then determine the height of the building.
28
Geometry
Chapter 11.2 Justifying Similar Triangles Worksheet 7
Name
Determine whether each pair of triangles is similar. If the triangles are similar, justify your answer by using
SSS, SAS, and AA. Make sure you have work to support your answer.
1.
Q
Yes
7
R
P
15
M
 _________ ~  _________
N
30
by ____________
21
O
45
2.
Yes
14
7
14
U
3
by ____________
S
E
3.
Yes
3
30˚
G
 _________ ~  _________
V
6
7
No
T
R
Q
No
10
No
 _________ ~  _________
F
5
I
by ____________
15
30˚
A
9
J
29
4.
Yes
R
No
 _________ ~  _________
120˚
by ____________
20˚
T
S
K
J
120˚
40˚
L
R
5.
Yes
30
No
 _________ ~  _________
42
by ____________
S
T
20
A
12.6
C
10.5
6
B
30
Geometry G
Sec 9.4 Notes
Name_________________________
Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then
the triangle formed is similar to the original triangle.
If BE || CD , then ABE ~ ACD
A
Let’s see why this is true.
If BE || CD , then the corresponding angles which
are congruent are:
B
E
_____  ________ and _____  ________.
C
By AA,  _________ ~  _________.
D
Examples
Complete the proportions for the given diagram.
a.
c.
MI MT

ML
LE

b.
L
MI

ML EL
I
MT
ME
T
M
E
We can use these proportions to solve for the missing sides of similar triangles..
1.
D
6
B
2.
x
10
A
8
E
C
2
3
4
x
31
Theorem: If a line is parallel to one side of a triangle and intersects the other two sides, then
it separates the sides into segments of proportional lengths.
(Also known as the Side-Splitter Theorem.)
A
If BE || CD , then
B
E
D
C
AB AE

.
BC ED
If you need to find either BE or CD, you still need
to use similar triangles. You CANNOT use the
Side-Splitter Theorem to find these two sides since
they are not “split” sides.
Examples
Write and solve proportions to solve for each variable.
1.
8
2.
x
x+5
10
5
8
6
A
3.
2x + 7
7
6
G
E
x+2
x
2
C
F
4.
4
B
x
y
D
H
40
14
K
y
6
J
32
Geometry
Chapter 11.6 Proportional Segments between Parallel Lines
Directions: Find the value each variable in the diagrams.
1.
J
7
9
A
E
10
a
18
M
2.
I
b
A
20
M
15
12
H
w
40
I
p
S
3.
12
h
33
34
Practice 9.4
Solve for x for each problem.
1]
2]
A
6
9
x
E
2
11
F
x
C
5
J
10
D
4]
L
x
H
12
B
3]
G
R
7
1
6
K
5
3
x
Q
4
O
I
N
5
M
P
4
T
S
35
5]
E x
8
A
6]
D
G
F
12
11
B
2
C
15
J
K
7]
H
3
I
12
9
6
x
8]
5
Q
L
7
x
12
9
R
O
4
7
N
x
M
P
T
S
13
36
Name_________________________________________ Date___________ Hour_______
Sect 9.5 – Triangle Midsegments
Use centimeters or degrees to find the measures of the following…
R
SR = ________
RN = ________
P
I
SN = ________
SP = ________
S
N
PR = ________
S = _________
N = ________
RI = ________
RPI = _______
 PIR = ______
IN = ________
R = _________
PI = ________
Notice anything???
Fill in the measures of all of the sides and angles of the triangle below. Did the same thing
occur as above??
R
E
B
A
K
37
Midsegment for Triangles
A segment whose endpoints are the midpoints of two sides of a triangle is parallel to
the third side and half its length.
X
1
MN  YZ
2
N
M
Y
Z
THEOREM: The __________________ of a triangle is __________the length
of the third side and is ________________ to it.
Examples:
1) In the triangle given, A, B, and C are midpoints of the sides of TUV . If TU=12. UV=16
and TV=20…
a) Find AB, BC, and AC
b) Name the three pairs of parallel segments
T
A
V
B
U
38
2) D is the midpoint of AC and E is the midpoint of BC .
A
D
B
E
C
a. If AD is 8 and AB is 12 find AC, DC, and DE
AC_________
DC___________ DE___________
b. If m CDE  98 , and DE is 17.9 Find mCAB and AB
mCAB __________ AB____________
c.
If m ABC  43 and AD is 13 and BC is 27, Find mDEC , BE and AC
mDEC _________ BE_________ AC___________
39
40
Geometry G
Sec 9.6 Notes
Name_________________________
Proportional Parts and Parallel Lines
Remember the Side-Splitter Theorem?
Theorem: If a line is parallel to one side of a triangle and intersects the other
two sides, it divides those two sides proportionally.
A
B
C
Given: BE || CD
E
AB AE

BC ED
Prove:
D
What happens if there are more than two parallel lines?
Theorem: If three or more parallel lines intersect two transversals, the parallel lines divide
the transversals proportionally.
A
B
Given: AB || CD || EF
AC BD

Conclusion:
CE DF
D
C
F
E
Examples:
1. Complete each proportion.
a.
b.
AB

BC EF
AB DE

AC
A
D
B
C
E
F
DF
BC EF
Write and solve a proportion to find the value of x.
c.

41
2.
3
2
x
14
3.
4
x
10
4.
5
6
x
24
10
5.
x-4
x-1
8
12
42
Name________________________________________ Date____________ Hour_______
Sect. 9.7 – Perimeters and Similarity
A
D
6
9
F
B
1)
C
12
8
E
Use the Pythagorean Theorem to find AC and DE.
AC = _____________
DE = ____________
2) Find the following ratios.
AB

DF
BC

EF
CA

ED
3) Are the triangles similar? YES or NO
If YES, name the similarity correspondence. _________~__________ by ________
4) Perimeter of ABC = ______________
5) Find the ratio of
Perimeter of DFE = _______________
perimeter ABC

perimeter DFE
6) Compare the ratios of part 2 and part 5. What do you notice??
43
Let’s try another pair of shapes.
B
C
X
15
A
Are the shapes similar?
Y
6
25
W
D
Perimeter of ABCD = _______
10
What is the similarity ratio?
Z
perimeterABCD
 ______
perimeterWXYZ
Perimeter WXYZ = _______
How does the similarity ratio compare to the ratio of perimeters?
If two triangles are similar, then the measures of the corresponding
perimeters are proportional to the measure to the corresponding sides.
If HIJ ~  LMN, then
perimeter HIJ HI
IJ HJ



perimeter LMN LM MN LN
The perimeter of GEO is 27 and GEO ~ MAT. Use ratios to find the value of each
variable.
M
12
8
A
T
16
x
E
O
y
z
G
44
The ratio found by comparing the measures of corresponding sides of similar triangles is called
the _______________________________ or the ______________________________
Find the scale factor for each pair of similar triangles.
S
A
O 4
1)
10
T
5
6
12
H
B
8
2)
U
18
32
20
C
24
15
X
24
O
B
M
BAM to HOT =
CUB to SOX =
HOT to BAM =
SOX to CUB =
The perimeter of MDF is 84 feet. If MDF ~ KNG and the scale factor of MDF to
3
KNG is
, find the perimeter of KNG.
4
45
46
Geometry
Chapter 11 Review
Name
Write each ratio in lowest terms.
1. 21 in to 18 in
2. 105 inches : 35 feet
Tell whether each pair of ratios forms a proportion.
3.
5
10
and
15
30
4.
10
15
and
4
3
Solve for x.
5.
6 2

x 5
6.
9 36

4
z
5 t 8

3
18
7.
Determine whether the figures are similar. If so, what is the scale factor that transforms the figure on the left
to the figure on the right?
8.
9.
15
10
20
6
15
24
18
12
8
9
12
Yes
9
No
Scale Factor _____________
(left to right)
Yes
No
Scale Factor _____________
(left to right)
47
Use the grid provided below to draw a figure that is similar to the given figure, with the indicated scale factor.
10. Scale factor of 2
11. Scale factor of
1
3
12. Scale factor of 3
13. Given DEF ~ JHG , find x and y. Show your work.
E
H
x
.
18
15
G
y
J
D
16
20
F
14. Given FGH ~ WXY , find the s and the length FH. Show your work.
F
W
1
s-1
X
Y
G
2
6
H
48
Determine whether each pair of triangles is similar. If the triangles are similar, justify your
answer by using SSS~, SAS~, and AA~. Make sure you have work to support your answer.
15.
D
Yes
No
W
96
75
32
25
 __________________ ~  __________________
by ___________
O
16.
M
S
P
84
6
X
28
H
B
Yes
24
12
 __________________ ~  __________________
by ___________
T
W
8
17.
No
S
A
B
Q
120˚
Yes
No
32˚
 __________________~  ____________
28˚
C
S
120˚
R
by ___________
49
Solve for x in each of the diagrams. Show your work.
18.
19.
7
9
8
12
x
10
x
18
20.
20
12
15
x
21.
The measure of the angles of a triangle are in a ratio of 2 : 3 : 7. Find the number of degrees
in the largest angle of the triangle.
22.
The shadow of a 12-foot tree is 18 feet long at the same time the shadow of a boy is 6 feet l
long. How tall is the boy?
50
23.
A pile of kick boards is 4ft. 4 inches tall and is 6 feet away from a sunbather. At 3:00 a nearby
8-foot lifeguard station casts a 14 foot shadow, will the sunbather have to move out of the shade of
the pile at 3:00?
Determine if the figures are similar. If the figures are similar, what is the scale factor that transforms the
figure on the left to the figure on the right? (Assume that if a pair of angles appears congruent then they are
congruent.)
24. Yes
No
25. Yes
Scale Factor ___________
No
Scale Factor ___________
4
3
15
12
5
5
4
5
10
10
7
9
26. Yes
3
14
No
27. Yes
Scale Factor ___________
No
Scale Factor ___________
16
8
10
2
2
6
16
7
3
3
4
4
10
6
4
5
5
7
51