Proportions and Similarity

1
Proportions and Similarity

• 9.1 Using Ratios and Proportions

• 9.2 Similar Polygons

• 9.3 Similar Triangles

• 9.4 Proportional Parts and Triangles

• 9.5 Triangles and Parallel Lines

• 9.6 Proportional Parts and Parallel Lines

• 9.7 Perimeters and Similarity

2
Using Ratios and Proportions
What You'll Learn
You will learn to use ratios and proportions to
solve problems.
1) ratio 2) proportion 3) cross products 4)
extremes 5) means
3
Using Ratios and Proportions
In 2000, about 180 million tons of solid waste
was created in the United States. The paper made
up about 72 million tons of this waste.
The ratio of paper waste to total waste is 72 to
180.
This ratio can be written in the following ways.
72 to 180
72180
72 180
Definition of Ratio A ratio is a comparison of two numbers by division.
a to b
ab
a b
where b ? 0
4
Using Ratios and Proportions
proportion
A __________ is an equation that shows two
equivalent ratios.
Every proportion has two cross products.
In the proportion to the right, the terms 20 and
3 are called the extremes,
and the terms 30 and 2 are called the means.

20(3)
30(2)
The cross products are 20(3) and 30(2).
60 60
equal
The cross products are always _____ in a
proportion.
5
Using Ratios and Proportions
Theorem 9-1 Property of Proportions For any numbers a and c and any nonzero numbers b and d,
Likewise,
6
Using Ratios and Proportions
Solve each proportion

30(6)

(30 x)2
3(x)
15(2x)
3x 60 2x
30x 180
x 6
5x 60
x 12
7
Using Ratios and Proportions
The gear ratio is the number of teeth on the
driving gear to the number of teeth on the driven
gear.
If the gear ratio is 52 and the driving
gear has 35 teeth, how many teeth does the
driven gear have?
5
35
x
2
5x 70
The driven gear has 14 teeth.
8
Using Ratios and Proportions
End of Section 9.1
9
Similar Polygons
What You'll Learn
You will learn to identify similar polygons.
1) polygons 2) sides 3) similar polygons 4)
scale drawing
10
Similar Polygons
A polygon is a ______ figure in a plane formed by
segments called sides.
closed
It is a general term used to describe a geometric
figure with at least three sides.
Polygons that are the same shape but not
necessarily the same size are called
______________.
similar polygons
The symbol is used to show that two figures
are similar.
?ABC ?DEF
11
Similar Polygons
Definition of Similar Polygons Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are ___________.
proportional
Polygon ABCD polygon EFGH
12
Similar Polygons
Determine if the polygons are similar. Justify
your answer.
6
4
5
7
1) Are corresponding angles are _________.
congruent
2) Are corresponding sides ___________.
proportional
0.66 0.71
The polygons are NOT similar!
13
Similar Polygons
Find the values of x and y if ?RST ?JKL
6
4
y 2
7
4(y 2) 42
4y 8 42
5
4
4y 34
x
7
4x 35
14
Similar Polygons
Scale drawings are often used to represent
something that is too large or too small to be
drawn at actual size.
Contractors use scale drawings to represent the
floorplan of a house.
Use proportions to find the actual dimensions of
the kitchen.
1.25 in.
.75 in.
1 in
1 in
L ft.
w ft.
16 ft
16 ft
(16)(1.25) w
(16)(.75) L
20 w
12 L
width is 20 ft.
length is 12 ft.
15
Similar Polygons
End of Section 9.2
16
Similar Triangles
What You'll Learn
You will learn to use AA, SSS, and SAS similarity
tests for triangles.
Nothing New!
17
Similar Triangles
Some of the triangles are similar, as shown below.
The Bank of China building in Hong Kong is one of
the ten tallest buildings in the world.
Designed by American architect I.M. Pei, the
outside of the 70-story building is sectioned
into triangles which are meant to resemble the
trunk of a bamboo plant.
18
Similar Triangles
In previous lessons, you learned several basic
tests for determining whether two triangles are
congruent. Recall that each congruence test
involves only three corresponding parts of each
triangle.
Likewise, there are tests for similarity that
will not involve all the parts of each triangle.
Postulate 9-1 AA Similarity If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are ______.
similar
C
F
D
A
E
B
If ?A ?D and ?B ?E, then ?ABC ?DEF
19
Similar Triangles
Two other tests are used to determine whether two
triangles are similar.
proportional
Theorem 9-2 SSS Similarity If the measures of the sides of a triangle are ___________ to the measures of the corresponding sides of another triangle, then the triangles are similar.
C
6
F
2
3
1
A
E
D
B
4
8
then the triangles are similar
then ?ABC ?DEF
20
Similar Triangles
proportional
Theorem 9-3 SAS Similarity If the measures of two sides of a triangle are ___________ to the measures of two corresponding sides of another triangle and their included angles are congruent, then the triangles are similar.
C
F
2
1
D
A
E
B
4
8
then ?ABC ?DEF
21
Similar Triangles
Determine whether the triangles are similar.
If so, tell which similarity test is used and
complete the statement.
6
10
14
, the triangles are similar by SSS similarity.
Since


15
21
9
JMP
Therefore, ?GHK ?
22
Similar Triangles
Fransisco needs to know the trees height. The
trees shadow is 18 feet longat the same time
that his shadow is 4 feet long.
If Fransisco is 6 feet tall, how tall is the tree?
1) The suns rays form congruent angles with
the ground.
2) Both Fransisco and the tree form right
angles with the ground.
6
4
t
18
4t 108
t 27
6 ft.
The tree is 27 feet tall!
4 ft.
18 ft.
23
Similar Triangles
Slade is a surveyor.
To find the distance across Muddy Pond, he forms
similar triangles and measures distances as
shown.
What is the distance across Muddy Pond?
10
8

It is 36 meters across Muddy Pond!
x
45
10x 360
x 36
24
Similar Triangles
End of Section 9.3
25
Proportional Parts and Triangles
What You'll Learn
You will learn to identify and use the
relationships between proportional parts of
triangles.
Nothing New!
26
Proportional Parts and Triangles
In ?PQR,
Are ?PQR and ?PST, similar?
corresponding angles
?PST ? ?PQR
?P ? ?P
?PQR ?PST. Why? (What theorem /
postulate?)
S
T
AA Similarity (Postulate 9-1)
27
Proportional Parts and Triangles
parallel
Theorem 9-4 If a line is _______ to one side of a triangle, and intersects the other two sides, then the triangle formed is _______ to the original triangle.
similar
?ABC ?ADE.
28
Proportional Parts and Triangles
29
Proportional Parts and Triangles
Theorem 9-5 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
30
Proportional Parts and Triangles
31
Proportional Parts and Triangles
Jacob is a carpenter.
Needing to reinforce this roof rafter, he
must find the length of the brace.
4
x
4
10
10x 16
32
Proportional Parts and Triangles
End of Section 9.4
33
Triangles and Parallel Lines
What You'll Learn
You will learn to use proportions to determine
whether lines are parallel to sides of triangles.
Nothing New!
34
Triangles and Parallel Lines
You know that 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 (Theorem 9-5).
The converse of this theorem is also true.
Theorem 9-6 If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side.
35
Triangles and Parallel Lines
Theorem 9-7 If a segment joins the midpoints of two sides of a triangle, then it is parallel to the third side, and its measure equals ________ the measure of the third side.
one-half
36
Triangles and Parallel Lines
Use theorem 9 7 to find the length of segment
DE.
A
x
11
E
D
22
C
B
37
Triangles and Parallel Lines
A, B, and C are midpoints of the sides of ?MNP.
Complete each statement.
2) If BC 14, then MN ____
28
s
3) If m?MNP s, then m?BCP ___
4) If MP 18x, then AC __
9x
38
Triangles and Parallel Lines
A, B, and C are midpoints of the sides of ?DEF.
1) Find DE, EF, and FD.
14 10 16
2) Find the perimeter of ?ABC
20
3) Find the perimeter of ?DEF
40
4) Find the ratio of the perimeter of
?ABC to the perimeter of ?DEF.
2040
12
39
Triangles and Parallel Lines
ABCD is a quadrilateral.
They are parallel
Q2) What kind of figure is EFHG ?
Parallelogram
40
Triangles and Parallel Lines
End of Section 9.5
41
Proportional Parts and Parallel Lines
What You'll Learn
You will learn to identify and use the
relationships betweenparallel lines and
proportional parts.
Nothing New!
42
Proportional Parts and Parallel Lines
Hands-On
On your given paper, draw two (transversals)
lines intersecting the parallel lines.
Label the intersections of the transversals and
the parallel lines, as shown here.
Do the parallel lines divide the transversals
proportionally?
Yes
43
Proportional Parts and Parallel Lines
Theorem 9-8 If three or more parallel lines intersect two transversals,the lines divide the transversals proportionally.
If l m n
44
Proportional Parts and Parallel Lines
Find the value of x.
UV
GH
HJ
VW
12
15
18
x
12x 18(15)
12x 270
45
Proportional Parts and Parallel Lines
Theorem 9-9 If three or more parallel lines cut off congruent segments onone transversal, then they cut off congruent segments onevery transversal.
If l m n and
Then
46
Proportional Parts and Parallel Lines
Find the value of x.
10
A
B
10
Theorem 9 - 9
C
(x 3) (2x 2)
x 3 2x 2
(2x 2)
8
(x 3)
8
F
E
D
47
Proportional Parts and Parallel Lines
End of Section 9.6
48
Perimeters and Similarity
What You'll Learn
You will learn to identify and use proportional
relationships of similar triangles.
1) Scale Factor
49
Perimeters and Similarity
These right triangles are similar! Therefore,
the measures of their corresponding sides are
___________.
proportional
Pythagorean
Use the ____________ theoremto calculate the
length of the hypotenuse.
10
6
15
9
8
12
8
10
2
6
We know that



12
15
3
9
Is there a relationship between the measures of
the perimeters of the two triangles?
50
Perimeters and Similarity
Theorem 9-10 If two triangles are similar, then

the measures of the corresponding perimeters
are proportional to

the measures of the
corresponding sides.
If ?ABC ?DEF, then
51
Perimeters and Similarity
The perimeter of ?RST is 9 units, and ?RST
?MNP.
Find the value of each variable.
Theorem 9-10
The perimeter of ?MNP is 3 6 4.5
3y 12
3z 9
27 13.5x
Cross Products
52
Perimeters and Similarity
The ratio found by comparing the measures of
corresponding sides of similar triangles is
called the constant of proportionality or the
___________.
scale factor
If ?ABC ?DEF, then
The scale factor of ?ABC to ?DEF is
Each ratio is equivalent to
The scale factor of ?DEF to ?ABC is
53
Perimeters and Similarity
End of Section 9.7
54
Proportional Parts and Triangles
Hands-On
A
Step 1) On a piece of lined paper, pick a point on one of the lines and label it A. Use a straightedge and protractor to draw ?A so that m?A lt 90 and only the vertex lies on the line.
Step 2) Extend one side of ?A down four lines. Label this point ?E. Do the same for the other side of ?A. Label this point ?I. Now connect points E and I to form ?AEI.
Step 3) Label the points where the horizontal lines intersect segment AG (B through D). Label the points where the horizontal lines intersect segment AI (F through H).
B
F
C
G
D
H
E
I
What can you conclude about the lines through the
sides of ?AEI and parallel to segment EI?
This activity suggests Theorem 9-5.