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Proportions and Similar Triangles
8.6
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Theorem 8.4 Triangle Proportionality Theorem
If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally.
Q
S
T
U
R.US
RU
TQ
RTThen
,QS toparallel is TU If
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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
S
T
U
R.QS toparallel is TUThen
,US
RU
TQ
RT If
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Example 1
In the diagram ll , BD=8,DC=4, and AE=12. What is the length of ?
AB ED
EC
C
B A
D E
4
8 12
AE
EC
BD
DC
128
4 EC EC
8
)12(4
EC6
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Example 2
Given the diagram, determine whether ll .MN GH
G
HL
M
N
21
56
16
48
3
8
21
56
MG
LM
1
3
16
48
NH
LN
1
3
3
8
.GH toparallelnot is NM
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Find x.
x
21=
40
36
36x = 840 36 36
x = 23⅓
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Find x.
6+
5=
2
xx2(x + 6) = 5x
2x + 12 = 5x-2x -2x
12 = 3x3 34 = x
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Find x.
2.5(8 – x) = 3.5x
20 – 2.5x = 3.5x +2.5x +2.5x
20 = 6x6 6 3⅓ = x
xx 8
5.3=
5.2
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Find x.
3x = 31.5 3 3
x = 10.5
3
7=
5.4
x
7
x
3
4.5
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Find x.
6x
10
x + 5
5+=
10
6
x
x
6(x + 5) = 10x6x + 30 = 10x-6x -6x
30 = 4x 4 4 7.5 = x
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Find x.
12.8x = 115.212.8 12.8
x = 9
8.12
6.9=
12
x
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Find x.
24x = 24024 24
x = 10 in.
24
15=
16
x
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Find x.
12x = 7212 12
x = 6 ft.
12
9=
8
x
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Theorem 8.6
If three parallel lines intersect two transversals, then they divide the transversals proportionally.
V
XZ
YWU
r s t
l
m.
WY
UWthen
t,s,r,intersect m and l and
parallel, all are t,and s, r, If
XZ
VX
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Example
?TU oflength theis What 11.ST and 15,QR
9,PQ and 3,21 diagram In the
1
2
3
S
T
UR
Q
P TU
ST
QR
PQ
TU
11
15
9
11 15 TU 9
3
118or
3
55TU
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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 two sides.
D
C
A
B
X
.AD
then
ACB, bisects CD If
CB
CA
DB
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Example
14BC .DC oflength thefind tolengths
sidegiven the UseDAB.CAD diagram, In the
DC
BD
AC
AB
x
x
14
15
9
9
15
A B
C
D
x
14-x21024
152109
)14(159
x
xx
xx
75.8x