8.6 Proportions & Similar Triangles

Objectives/Assignments

Use proportionality theorems to calculate segment lengths

Assignment: 2-34 even

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

S

R

T

U

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, thenRT

TQ

RUUS

=

Theorems 8.5 Converse of the Triangle Proportionality Theorem

Q

S

R

T

U

If a line divides two sides of a triangle proportionally, then it is parallel to the third side.

RT

TQ

RUUS

=If , 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?

128

4

C

B A

D E

Step:

DC EC

BD AE

4 EC

8 12

4(12)

8

6 = EC

ReasonTriangle

Proportionality Thm.

SubstituteMultiply each side

by 12.Simplify.

128

4

C

B A

D E

=

=

= EC

So, the length of EC is 6.

Ex. 2: Determining Parallels

Given the diagram, determine whether MN ║ GH.

21

1648

56

L

G

H

M

N

LMMG

56

21=

8

3=

LN

NH

48

16=

3

1=

8

3

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

UWWY

VX

XZ=

l

m

s

Z

YW

XV

U

rt

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.

If CD bisects ACB, then AD

DB

CA

CB=

D

C

A

B

Ex. 3: Using Proportionality Theorems

In the diagram 1 2 3, and PQ = 9, QR = 15, and ST = 11. What is the length of TU?

11

15

9

3

2

1

S

T

UR

Q

P

SOLUTION: Because corresponding angles are congruent, the lines are parallel and you can use Theorem 8.6

PQ

QR

ST

TU=

9

15

11

TU=

9 ● TU = 15 ● 11 Cross Product property

15(11)

9

55

3=TU =

Parallel lines divide transversals proportionally.

Substitute

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

In the diagram, CAD DAB. Use the given side lengths to find the length of DC.

15

9

14D

A

C

B

15

9

14D

A

C

B

Solution:

Since AD is an angle bisector of CAB, you can apply Theorem 8.7. Let x = DC. Then BD = 14 – x.AB

AC

BDDC

=

9

15

14-X

X=

Apply Thm. 8.7

Substitute.

Ex. 4 Continued . . .

9 ● x = 15 (14 – x)

9x = 210 – 15x

24x= 210

x= 35/4 = 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.

y

x

37.5

13.5

9

7.5

J

M N

KL

Solution

To find the value of x, you can set up a proportion.

y

x

37.5

13.5

9

7.5

J

M N

KL

9

13.5

37.5 - x

x=

13.5(37.5 – x) = 9x

506.25 – 13.5x = 9x

506.25 = 22.5 x

22.5 = x

Write the proportion

Cross product property

Distributive property

Add 13.5x to each side.

Divide each side by 22.5Since KL ║MN, ∆JKL ~ ∆JMN and JK

JM

KL

MN=

Solution

To find the value of y, you can set up a proportion.

y

x

37.5

13.5

9

7.5

J

M N

KL

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.