Indirect MeasurementIndirect Measurementand Additionaland Additional
Similarity Theorems 8.5Similarity Theorems 8.5
• Learn the triangle angle bisector theorem.
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• Learn the proportional altitudes theorem.
• Learn the proportional medians theorem.
Proportional Angle Bisectors TheoremProportional Angle Bisectors Theorem
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Proportional Altitudes TheoremProportional Altitudes Theorem
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Proportional Medians TheoremProportional Medians Theorem
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Proportional Angle Bisector Theorem
Find PS and SR.
40(x – 2) = 32(x + 5)
40x – 80 = 32x + 160
8x = 240
x = 30
PS = x – 2 SR = x + 5
RS
RQ
PS
PQ
5x
40
2x
32
PS = 28 SR = 35
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A. A
B. B
C. C
D. D
The drawing of the table below has legs AE and CG. AC measures 12 inches and GE measures 36 inches. If BD measures 7 inches, what is the measure of DF and what is the height of the table?
Proportional Altitudes Theorem
12x = 252
x = 21
DF
GE
BD
AC
x
36
7
12
Therefore, DF = 21, and the table is 28 inches tall. homeworkhomework
Proportional Medians TheoremIn the figure, EFD ~ JKI. EG is a median of EFG and JL is a median of JKL. Find JL if EF = 36, EG = 18, and JK = 56.
1008 = 36x
x = 28
JK
EF
JL
EG
56
36
x
18
Therefore, JL = 28.
3618
56
E
D G F
K
J
I L
x
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Use indirect measure to find the missing value.
26d = 800
40
d
26
20
inches13
1030d
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Use indirect measure to find the missing value.
10x = 360
45
x
10
8
x = 36
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Use indirect measure to find the missing value.
4t = 108
18
t
4
6
x = 27 feet
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Use indirect measure to find the missing value.
4x = 55
10
x
4
5.5
x = 13.75 feet
SR = 25, ST = 15
Use the Triangle Angle Bisector Theorem find ST and SR
7.5x + 37.5 = 12.5x – 62.5
–5x = –100
TV
ST
RV
RS
5.7
5x
5.12
5x
x = 20
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EXAMPLE 4
In the diagram, QPR RPS. Use the given side lengths to find the length of RS .
7x = 195 – 13x
x = 9.75
SR
PS
QR
PQ
x
13
x15
7
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Use the Triangle Angle Bisector Theorem find x.
42x2 = 50x2 – 20x
8x2 – 20x = 0
x6
4x10
x5
x7
4x(2x – 5) = 0
x = 0 or x = 2.5
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4x = 36 – 6x
x = 3.6
x6
4
x
6
Use the Triangle Angle Bisector Theorem find x.
10x = 36
AssignmentAssignment
8.5 Indirect 8.5 Indirect MeasurmentMeasurment