indirect measurement and additional similarity theorems 8.5

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