Transcript Slide 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
Using Ratios and Proportions
You will learn to use ratios and proportions to solve problems.
1) ratio
2) proportion
3) cross products
4) extremes
5) means
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
72:180
72
72 ÷ 180
180
A ratio is a comparison of two numbers by division.
Definition of
Ratio
a to b
a:b
a
b
a÷b
where b  0
Using Ratios and Proportions
proportion is an equation that shows two equivalent ratios.
A __________
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
30

2
3
30(2) = 20(3)
The cross products are 20(3) and 30(2).
equal
The cross products are always _____
in a proportion.
60 = 60
Using Ratios and Proportions
For any numbers a and c and any nonzero numbers b and d,
if
a

b
Theorem 9-1
Property of
Proportions
Likewise,
c
, then
ad  bc
d
if ad  bc , then
a
b
if
5
10
If

1
, then

c
.
d
5 2   10 1 
2
5 2   10 1 
then
5
10

1
2
Using Ratios and Proportions
Solve each proportion:
6
2x

15
30  x
30
x

3
2
15(2x) = 30(6)
3(x) = (30 – x)2
30x = 180
3x = 60 – 2x
x=6
5x = 60
x = 12
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.
Driving gear
If the gear ratio is 5:2 and the driving gear
has 35 teeth, how many teeth does the
driven gear have?
given
ratio
driving gear
5
driven gear
2
=
=
Driven gear
equivalent
ratio
35
driving gear
x
driven gear
5x = 70
x = 14
The driven gear has 14 teeth.
Using Ratios and Proportions
Similar Polygons
You will learn to identify similar polygons.
1) polygons
2) sides
3) similar polygons
4) scale drawing
Similar Polygons
closed figure in a plane formed by segments called sides.
A polygon is a ______
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
similar polygons
called ______________.
The symbol ~ is used to show that two figures are similar.
D
ΔABC is similar to ΔDEF
A
C
B
ΔABC ~ ΔDEF
F
E
Similar Polygons
Two polygons are similar if and only if their corresponding
angles are congruent and the measures of their corresponding
proportional
sides are ___________.
D
C
G
H
Definition of
Similar
Polygons
A
E
B
F
A  E,
B  F
C  G,
D  H
AB
EF

BC
FG

CD
GH

and
DA
HE
Polygon ABCD ~ polygon EFGH
Similar Polygons
Determine if the polygons are similar. Justify your answer.
6
4
4
5
7
5
7
4
6
congruent
1) Are corresponding angles are _________.
proportional
2) Are corresponding sides ___________.
=
0.66 = 0.71
The polygons are NOT similar!
Similar Polygons
Find the values of x and y if ΔRST ~ ΔJKL
R
4
7
T
6
J
S
6
=
7
y + 2
4(y + 2) = 42
4y + 8 = 42
4y = 34
y 8
5
4
Write the proportion that
can be solved for y.
1
x
Write the proportion that
can be solved for x.
L
4
7
5
=
x
4x = 35
2
x 8
3
4
y + 2
K
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.
.75 in.
length
width
1 in
16 ft
=
1.25 in.
1 in
w ft.
16 ft
=
.75 in.
L ft.
(16)(1.25) = w
(16)(.75) = L
20 = w
12 = L
width is 20 ft.
length is 12 ft.
1.25 in.
1 in.
1.25 in.
.5 in.
Dining
Room
Kitchen
Utility
Room
Living
Room
Garage
Scale: 1 in. = 16 ft.
Similar Polygons
Similar Triangles
You will learn to use AA, SSS, and SAS similarity tests for
triangles.
Nothing New!
Similar Triangles
Somebuilding
of the
triangles
asofshown
below.
The
Designed
Bank of
by China
American
architect
in Hong
I.M.are
Kong
Pei,similar,
the
is one
outside
the
often
the
tallest
70-story
buildings
buildingin
the
is sectioned
world. into triangles which are meant to resemble the trunk of a bamboo
plant.
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.
If two angles of one triangle are congruent to two
corresponding angles of another triangle, then the triangles
similar
are ______.
Postulate
9-1
AA Similarity
C
F
A
B
D
E
If A ≈ D and B ≈ E, then ΔABC ~ ΔDEF
Similar Triangles
Two other tests are used to determine whether two triangles are similar.
proportional
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
Theorem 9-2
6
2
SSS
Similarity
A
1
8
If
B
D
F
3
4
E
8AB 2 AC6
CB
  
 then then
the triangles
are similar
ΔABC ~ ΔDEF
4DE 1 DF3
FE
Similar Triangles
proportional
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
Theorem 9-3
SAS
Similarity
2
1
A
B
8
If
AB 8 AC
2
If

DE 4 DF
1
and
F
D
A  D
then ΔABC ~ ΔDEF
4
E
Similar Triangles
Determine whether the triangles are similar.
is used and complete the statement.
14
G
6
K
If so, tell which similarity test
J
9
21
10
H
M
15
P
Since
6
9
=
10
15
=
14
21
Therefore, ΔGHK ~ Δ JMP
, the triangles are similar by SSS similarity.
Similar Triangles
Fransisco needs to know the tree’s height. The tree’s shadow is 18 feet long
at the same time that his shadow is 4 feet long.
If Fransisco is 6 feet tall, how tall is the tree?
1) The sun’s rays form congruent angles with the
ground.
2) Both Fransisco and the tree form right
angles with the ground.
4
=
18
6
t
4t = 108
t
= 27
The tree is
27 feet tall!
6 ft.
4 ft.
18 ft.
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
45
=
8
x
10x = 360
x = 36
45 m
x
8 m
10 m
It is 36 meters across Muddy Pond!
Similar Triangles
Proportional Parts and Triangles
You will learn to identify and use the relationships between
proportional parts of triangles.
Nothing New!
Proportional Parts and Triangles
In ΔPQR, ST || QR
and ST
intersects the other two sides of ΔPQR.
Are ΔPQR and ΔPST, similar?
P
PST  PQR
corresponding angles
P  P
ΔPQR ~ ΔPST.
S
Why? (What theorem / postulate?)
AA Similarity (Postulate 9-1)
Q
T
R
Proportional Parts and Triangles
parallel to one side of a triangle, and intersects the
If a line is _______
similar to the
other two sides, then the triangle formed is _______
original triangle.
A
Theorem 9-4
B
C
E
D
If BC || DE ,
then ΔABC ~ ΔADE.
Proportional Parts and Triangles
Since VW || RT , ΔSVW ~ ΔSRT.
Complete the proportion:
ST

SW
?
ST
SR
SW

R
SR
SV
V
S
W
T
Proportional Parts and Triangles
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
__________________.
A
B
Theorem 9-5
C
D
E
If BC || DE , then
AB
BD

AC
CE
Proportional Parts and Triangles
In the figure,
GH || BC .
Find the value of x
A
3
3
x

G
4
x
x5
4 x  3 x  5 
4 x  3 x  15
x  15
B
4
5
H
x + 5
C
Proportional Parts and Triangles
Jacob is a carpenter.
Needing to reinforce this roof rafter, he must
find the length of the brace.
4
10
Brace
x
=
x
4
4 ft
10x = 16
x =
1
3
5
ft
10 ft
6 ft
4 ft
Proportional Parts and Triangles
Triangles and Parallel Lines
You will learn to use proportions to determine whether lines
are parallel to sides of triangles.
Nothing New!
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.
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.
A
6
4
C
B
Theorem 9-6
9
6
D
E
If
4
6

6
9
, then
BC || DE
Triangles and Parallel Lines
If a segment joins the midpoints of two sides of a triangle, then
it is parallel to the third side, and its measure equals
one-half the measure of the third side.
________
A
D
x
E
Theorem 9-7
B
2x
C
If D is the midpoint
of AB , and
E is the midpoint
of AC , then
DE || BC , and
DE 
1
2
BC
Triangles and Parallel Lines
Use theorem 9 – 7 to find the length of segment DE.
A
x 
1
2
8
5
22 
D
x
11
E
8
5
x  11
B
22
C
If D is the midpoint
of AB , and
E is the midpoint
of AC , then
DE || BC , and
DE 
1
2
BC
Triangles and Parallel Lines
A, B, and C are midpoints of the sides of ΔMNP.
M
Complete each statement.
1) MP || ____
AC
A
B
2) If BC = 14, then MN = ____
28
N
s
3) If mMNP = s, then mBCP = ___
4) If MP = 18x, then AC = 9x
__
C
P
Triangles and Parallel Lines
A, B, and C are midpoints of the sides of ΔDEF.
E
1) Find DE, EF, and FD. 14; 10; 16
2) Find the perimeter of ΔABC 20
A
3) Find the perimeter of ΔDEF 40
5
D
4) Find the ratio of the perimeter of
ΔABC to the perimeter of ΔDEF.
8
20:40 = 1:2
B
7
C
F
Triangles and Parallel Lines
ABCD is a quadrilateral.
D
F
E is the midpoint of AD
C
F is the midpoint of DC
E
H
H is the midpoint of CB
G is the midpoint of BA
A
Q1) What can you say about EF and GH ?
G
They are parallel
(Hint: Draw diagonal AC .)
Q2) What kind of figure is EFHG ?
Parallelogram
B
Triangles and Parallel Lines
Proportional Parts and Parallel Lines
You will learn to identify and use the relationships between
parallel lines and proportional parts.
Nothing New!
Proportional Parts and Parallel Lines
On your given paper,
draw two (transversals)
lines intersecting the parallel lines.
D
A
E
B
Label the intersections of the
transversals and the parallel lines,
as shown here.
F
C
Measure AB, BC, DE, and EF.
Calculate each set of ratios:
AB
BC
,
DE
AB
EF
AC
,
DE
DF
Do the parallel lines divide the transversals proportionally?
Yes
Proportional Parts and Parallel Lines
If three or more parallel lines intersect two transversals,
the lines divide the transversals proportionally.
A
D
B
Theorem 9-8
l
E
m
C
F
If l || m || n
Then
AB
BC
=
DE
EF
,
AB
AC
=
DE
DF
, and
BC
AC
=
EF
DF
n
Proportional Parts and Parallel Lines
Find the value of x.
GH
HJ
12
18
=
=
UV
G
U
12
15
H
VW
15
V
18
x
12x = 270
x = 22
1
2
b
x
J
12x = 18(15)
a
W
c
Proportional Parts and Parallel Lines
If three or more parallel lines cut off congruent segments on
one transversal, then they cut off congruent segments on
every transversal.
A
B
D
l
E
m
Theorem 9-9
C
If l || m || n
F
and AB  BC,
Then DE  EF.
n
Proportional Parts and Parallel Lines
Find the value of x.
A
Since AB  BC,
DE  EF
10
B
10
Theorem 9 - 9
C
(x + 3) = (2x – 2)
x + 3 = 2x – 2
5=x
(x 8+3)
D
(2x 8– 2)
E
F
Proportional Parts and Parallel Lines
Perimeters and Similarity
You will learn to identify and use proportional relationships of
similar triangles.
1) Scale Factor
Perimeters and Similarity
These right triangles are similar!
proportional
Therefore, the measures of their corresponding sides are ___________.
Pythagorean theorem
Use the ____________
to calculate the length of the
hypotenuse.
c
2
2
 a b
10
6
2
9
8
15
12
We know that
10
2
8
6
=
=
=
15
3
12
9
Is there a relationship between the measures of the perimeters of the two
triangles?
perimeter of small Δ
perimeter of large Δ
=
6 + 8 + 10
9 + 12 + 15
=
24
36
=
2
3
Perimeters and Similarity
If two triangles are similar, then the measures of the
corresponding perimeters are proportional to the measures
of the corresponding sides.
D
A
Theorem
9-10
C
B
F
E
If ΔABC ~ ΔDEF, then
perimeter of ΔABC
perimeter of ΔDEF
=
CA
BC
AB
=
=
FD
EF
DE
Perimeters and Similarity
The perimeter of ΔRST is 9 units, and ΔRST ~ ΔMNP.
Find the value of each variable.
perimeter of ΔMNP
MN
=
RS
perimeter of ΔRST
3
x
=
13.5
9
27 = 13.5x
x = 2
NP
MN
=
Theorem 9-10 ST
RS
6
3
=
The perimeter of ΔMNP 2is 3 +y 6 + 4.5
Cross Products
3y = 12
y = 4
PM
MN
=
TR
RS
4.5
3
=
z
2
3z = 9
z = 3
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
D
A
5
3
B
7
C
E
If ΔABC ~ ΔDEF, then
or
10
6
F
14
CA
BC
AB
=
=
FD
EF
DE
7
5
3
=
=
6
14
10
Each ratio is equivalent to
1
2
The scale factor of ΔABC to ΔDEF is
1
2
The scale factor of ΔDEF to ΔABC is
2
1
Perimeters and Similarity
Proportional Parts and Triangles
A
1)
2)
Step 3)
F
B
C
G
D
H
E
I
On a piece
of
lined
paper,
Extend
one
side
of A
down
Label
the
points
where
thepick
a
point
on lines
one
ofintersect
the lines
four
lines.
Label
this
point and
E.
horizontal
label
it A.
segment
AG (B
D).side
Do the
same
forthrough
the other
Use
a the
straightedge
andthe
of
A.
Label
thiswhere
point
I.
Label
points
protractor
to
draw
A
that
horizontal
lines
intersect
Now connect
points
E so
and
I to
mA
< 90AIand
only theH).
vertex
segment
(F through
form ΔAEI.
lies on the line.
What can you
conclude
about
the sides
Calculate
and
the, lines
following
: of ΔAEI and
Measure
AC
, compare
AD , AE
, AG
AH
, through
and ratios
AI
parallel to segment EI?
AC
, and
AG
AD
, and
This
Theorem
CE activity suggests
GI
AE 9-5.
AH
AI