8.6 Proportions & Similar Triangles Geometry Mrs. Spitz Spring 2005 Objectives/Assignments Use proportionality theorems to calculate segment lengths. To solve real-life problems, such as determining the dimensions of.
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Transcript 8.6 Proportions & Similar Triangles Geometry Mrs. Spitz Spring 2005 Objectives/Assignments Use proportionality theorems to calculate segment lengths. To solve real-life problems, such as determining the dimensions of.
8.6 Proportions & Similar
Triangles
Geometry
Mrs. Spitz
Spring 2005
Objectives/Assignments
Use proportionality theorems to
calculate segment lengths.
To solve real-life problems, such as
determining the dimensions of a piece
of land.
Assignment: pp. 502-503 #1-30
Use Proportionality Theorems
In this lesson, you will study four
proportionality theorems. Similar
triangles are used to prove each
theorem.
Theorems 8.4 Triangle
Proportionality Theorem
Q
If a line parallel to one side
of a triangle intersects the
other two sides, then it
divides the two side
proportionally.
If TU ║ QS, then
RT
TQ
=
RU
US
T
R
S
U
Theorems 8.5 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
T
R
S
If
RT
TQ
=
RU
US
U
, then TU ║ QS.
Ex. 1: Finding the length of a
segment
In the diagram AB ║ ED, BD = 8, DC =
4, and AE = 12. What is the length of
EC?
C
4
D
8
B
E
12
A
C
4
D
8
B
Step:
DC = EC
BD AE
4 EC
=
8 12
4(12)
= EC
8
6 = EC
So, the length of EC is 6.
E
12
A
Reason
Triangle Proportionality
Thm.
Substitute
Multiply each side by
12.
Simplify.
Ex. 2: Determining Parallels
Given the diagram, determine whether
MN ║ GH.
LM
G
MG
=
56
21
8
=
3
21
M
LN
NH
56
=
48
16
8
L
N
48
16
H
3
=
≠
3
1
3
1
MN is not parallel to GH.
Theorem 8.6
If three parallel lines intersect two transversals, then
they divide the transversals proportionally.
If r ║ s and s║ t and l and m intersect, r, s, and t,
then
UW
WY
=
VX
XZ
t
s
r
U
W
Y
m
V
X
Z
Theorem 8.7
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.
AD
CA
=
If CD bisects ACB, then DB CB
A
D
B
C
Ex. 3: Using Proportionality
Theorems
P
S
1
9
11
Q
T
2
15
R
U
3
In the diagram 1
2 3, and PQ =
9, QR = 15, and ST
= 11. What is the
length of TU?
SOLUTION: Because corresponding angles are
congruent, the lines are parallel and you can use
Theorem 8.6
PQ
QR
9
15
=
=
ST
TU
11
TU
Parallel lines divide transversals
proportionally.
Substitute
9 ● TU = 15 ● 11 Cross Product property
TU
15(11)
55
=
=
9
3
Divide each side by 9 and simplify.
So, the length of TU is 55/3 or 18 1/3.
Ex. 4: Using the
Proportionality Theorem
9
A
B
D
14
15
C
In the diagram,
CAD DAB. Use
the given side
lengths to find the
length of DC.
Solution:
9
A
B
D
14
15
Since AD is an angle bisector of
CAB, you can apply
Theorem 8.7. Let x = DC.
Then BD = 14 – x.
AB
AC
=
BD
DC
Apply Thm. 8.7
C
9
15
=
14-X
X
Substitute.
Ex. 4 Continued . . .
9 ● x = 15 (14 – x)
9x = 210 – 15x
24x= 210
x= 8.75
Cross product property
Distributive Property
Add 15x to each side
Divide each side by 24.
So, the length of DC is 8.75 units.
Use proportionality Theorems
in Real Life
Example 5: Finding the
length of a segment
Building Construction:
You are insulating your
attic, as shown. The
vertical 2 x 4 studs are
evenly spaced. Explain
why the diagonal cuts at
the tops of the strips of
insulation should have the
same length.
Use proportionality Theorems
in Real Life
Because the studs AD, BE
and CF are each vertical,
you know they are parallel
to each other. Using
Theorem 8.6, you can
conclude that
DE
EF
=
AB
BC
Because the studs are evenly spaced, you know that
DE = EF. So you can conclude that AB = BC, which
means that the diagonal cuts at the tops of the strips
have the same lengths.
Ex. 6: Finding Segment
Lengths
In the diagram KL ║
MN. Find the values
of the variables.
J
9
K
L
37.5
7.5
x
13.5
M
y
N
J
9
K
Solution
L
37.5
7.5
x
13.5
M
y
N
To find the value of x, you can set up a proportion.
9
13.5
=
37.5 - x
x
Write the proportion
13.5(37.5 – x) = 9x
Cross product property
506.25 – 13.5x = 9x
Distributive property
506.25 = 22.5 x
Add 13.5x to each side.
22.5 = x
Divide each side by 22.5
Since KL ║MN, ∆JKL ~ ∆JMN and JK
KL
JM
=
MN
J
9
K
Solution
L
37.5
7.5
x
13.5
M
y
N
To find the value of y, you can set up a proportion.
9
13.5 + 9
=
7.5
y
9y = 7.5(22.5)
y = 18.75
Write the proportion
Cross product property
Divide each side by 9.