Proportions & Similar Triangles

Objectives/Assignments

Use proportionality theorems to calculate segment lengths.

To solve real-life problems, such as determining the dimensions of a piece of land.

Use Proportionality Theorems

In this lesson, you will study four proportionality theorems. Similar triangles are used to prove each theorem.

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

=

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.

Proportional parts of // lines

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

Special segments 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.

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= 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.

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.