§ 9.1 Using Ratios and Proportions § 9.4 Proportional Parts and Triangles § 9.3 Similar Triangles § 9.2 Similar Polygons § 9.6 Proportional Parts

Proportions and SimilarityProportions and SimilarityProportions and SimilarityProportions and Similarity

§§ 9.1 Using Ratios and Proportions 9.1 Using Ratios and Proportions

§§ 9.4 Proportional Parts and Triangles 9.4 Proportional Parts and Triangles

§§ 9.3 Similar Triangles 9.3 Similar Triangles

§§ 9.2 Similar Polygons 9.2 Similar Polygons

§§ 9.6 Proportional Parts and Parallel Lines 9.6 Proportional Parts and Parallel Lines

§§ 9.5 Triangles and Parallel Lines 9.5 Triangles and Parallel Lines

§§ 9.7 Perimeters and Similarity 9.7 Perimeters and Similarity

Using Ratios and Proportions Using Ratios and Proportions

You will learn to use ratios and proportions to solve problems.

1) ratio

2) proportion

3) cross products

4) extremes

5) means


In 2000, about 180 million tons of solid waste was created in the United States.

The paper made up about 72 million tons of this waste.

The ratio of paper waste to total waste is 72 to 180.

This ratio can be written in the following ways.

72 to 180 72:180 72 ÷ 180180

72

Definition of

Ratio

A ratio is a comparison of two numbers by division.

a to b a:b a ÷ bb

awhere b 0


A __________ is an equation that shows two equivalent ratios. proportion

3

2

30

20

Every proportion has two cross products.

In the proportion to the right, the terms 20 and 3 are called the extremes,and the terms 30 and 2 are called the means.

The cross products are 20(3) and 30(2).

The cross products are always _____ in a proportion.

equal

30(2) = 20(3)

60 = 60


Theorem 9-1

Property of

Proportions

For any numbers a and c and any nonzero numbers b and d,

, ifd

c

b

a

Likewise, , if bc ad

then , if2

1

10

5 11025

2

1

10

5 then If 11025

bcad then

. thend

c

b

a


Solve each proportion:

30

15

2

6

x

15(2x) = 30(6)

30x = 180

x = 6

2

330

x

x

3(x) = (30 – x)2

3x = 60 – 2x

5x = 60

x = 12


The gear ratio is the number of teeth on the driving gear to the number of teeth onthe driven gear.

Driving gear

Driven gear

If the gear ratio is 5:2 and the driving gearhas 35 teeth, how many teeth does the driven gear have?

givenratio

equivalentratio=

5

2

35

x=driving gear

driven gear

driving gear

driven gear

5x = 70

x = 14 The driven gear has 14 teeth.


Similar Polygons Similar Polygons

You will learn to identify similar polygons.

1) polygons

2) sides

3) similar polygons

4) scale drawing


A polygon is a ______ figure in a plane formed by segments called sides. closed

It is a general term used to describe a geometric figure with at least three sides.

Polygons that are the same shape but not necessarily the same size are called ______________.similar polygons

The symbol ~ is used to show that two figures are similar.

ΔABC ~ ΔDEF

ΔABC is similar to ΔDEFA

BC

D

F E


Definition of

Similar

Polygons

Two polygons are similar if and only if their corresponding angles are congruent and the measures of their corresponding sides are ___________.proportional

CD

A B FE

GH

HE

DA

GH

CD

FG

BC

EF

AB

Polygon ABCD ~ polygon EFGH

HDGC

FBEA

,

,and


Determine if the polygons are similar. Justify your answer.

4

5

4

5 7

6

6

7

1) Are corresponding angles are _________.congruent

2) Are corresponding sides ___________.proportional

4

=

7

6

5

0.66 = 0.71 The polygons are NOT similar!


Find the values of x and y if ΔRST ~ ΔJKL

R

T SJ

L K

4 5

6

7x

y + 2

4=

7

5

x

Write the proportion thatcan be solved for x.

4x = 35

4

38x

4=

7

6

y + 2

Write the proportion thatcan be solved for y.

4(y + 2) = 42

2

18y

4y + 8 = 42

4y = 34

width length


Scale drawings are often used to represent something that is too large or too small to be drawn at actual size.

Contractors use scale drawings to represent the floorplan of a house.

DiningRoom

Kitchen

LivingRoom

Garage

UtilityRoom

1.25 in.

.75 in.

1 in. 1.25 in. .5 in.

Scale: 1 in. = 16 ft.

Use proportions to find the actualdimensions of the kitchen.

1 in=

16 ft

1.25 in.

w ft.

(16)(1.25) = w

20 = w

width is 20 ft.

1 in=

16 ft

.75 in.

L ft.

(16)(.75) = L

12 = L

length is 12 ft.


Similar Triangles Similar Triangles

You will learn to use AA, SSS, and SAS similarity tests fortriangles.

Nothing New!


The Bank of China building in Hong Kong is one of the ten tallest buildings inthe world. Designed by American architect I.M. Pei, the outside of the 70-story buildingis sectioned into triangles which are meant to resemble the trunk of a bambooplant.

Some of the triangles are similar, as shown below.


In previous lessons, you learned several basic tests for determining whethertwo triangles are congruent. Recall that each congruence test involves onlythree corresponding parts of each triangle.

Likewise, there are tests for similarity that will not involve all the parts ofeach triangle.

Postulate

9-1

AA Similarity

If two angles of one triangle are congruent to two corresponding angles of another triangle, then the triangles are ______.similar

C

A B

F

ED

If A ≈ D and B ≈ E, then ΔABC ~ ΔDEF


Theorem 9-2

SSS Similarity

If the measures of the sides of a triangle are ___________

to the measures of the corresponding sides of another triangle, then the triangles are similar.

Two other tests are used to determine whether two triangles are similar.

proportional

C

A B

F

ED

12 3

6

48

1

2

4

8 If

3

6 then the triangles are similarDF

AC

DE

AB If

FE

CB then ΔABC ~ ΔDEF


Theorem 9-3

SAS Similarity

If the measures of two sides of a triangle are ___________

to the measures of two corresponding sides of another triangle and their included angles are congruent, then the triangles are similar.

proportional

C

A B

F

ED

12

48

4

8 If D A and

then ΔABC ~ ΔDEF

DF

AC

DE

AB If

1

2


Determine whether the triangles are similar. If so, tell which similarity testis used and complete the statement.

G

H

K

M

P

J

6

14

10

9

15

21

15=

9

6 10 , the triangles are similar by SSS similarity.Since =21

14

Therefore, ΔGHK ~ Δ JMP


Fransisco needs to know the tree’s height. The tree’s shadow is 18 feet longat the same time that his shadow is 4 feet long.

If Fransisco is 6 feet tall, how tall is the tree?

1) The sun’s rays form congruent angles with the ground.

2) Both Fransisco and the tree form right angles with the ground.

6 ft.

4 ft. 18 ft.

4=

18

6

t

4t = 108

t = 27

The tree is 27 feet tall!


To find the distance across Muddy Pond, he forms similar triangles and measures distances as shown.

Slade is a surveyor.

8 m

10 m

45 m

xWhat is the distanceacross Muddy Pond?

x=

45

10 8

10x = 360

x = 36

It is 36 meters across Muddy Pond!


Proportional Parts and Triangles Proportional Parts and Triangles

You will learn to identify and use the relationships between proportional parts of triangles.

Nothing New!


In ΔPQR,

TS

RQ

P

Are ΔPQR and ΔPST, similar?

QRST || and intersects the other two sides of ΔPQR.ST

PST PQR corresponding angles

P P

ΔPQR ~ ΔPST. Why? (What theorem / postulate?)

AA Similarity (Postulate 9-1)


Theorem 9-4

If a line is _______ to one side of a triangle, and intersects the other two sides, then the triangle formed is _______ to the original triangle.

similarparallel

A

B C

D E

then ,|| DEBCIf ΔABC ~ ΔADE.


TWS

V

R

Complete the proportion:

?

SR

SW

ST

Since RTVW || , ΔSVW ~ ΔSRT.

SR

SW

STSV


Theorem 9-5

If a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of

__________________.proportional lengths

ED

CB

A

CE

AC

BD

ABDEBC then ,|| If


CB

HG

A

5

3

x + 5x

4

x

BCGH

of value the Find

.|| figure, the In

5

43

xx

534 xx

1534 xx

15x


10 ft

6 ft

4 ftBrace

Jacob is a carpenter.

Needing to reinforce this roof rafter, he mustfind the length of the brace.

4=

10

4 x

10x = 16

x = 1 3

5ft

4 ft

x


Triangles and Parallel Lines Triangles and Parallel Lines

You will learn to use proportions to determine whether linesare parallel to sides of triangles.

Nothing New!


You know that if a line is parallel to one side of a triangle and intersects the other two sides, then it separates the sides into segments of proportional lengths (Theorem 9-5).

The converse of this theorem is also true.

Theorem 9-6

If a line intersects two sides of a triangle and separates the sides into corresponding segments of proportional lengths, then the line is parallel to the third side.

ED

CB

A

9

6

6

4

then , If9

6

6

4 DEBC ||


Theorem 9-7

If a segment joins the midpoints of two sides of a triangle, then it is parallel to the third side, and its measure equals ________ the measure of the third side.one-half

ED

CB

A

and , of midpoint the is D If AB

2x

x

then , of midpoint the is E AC

and ,|| BCDE BCDE2

1


ED

CB

A

and , of midpoint the is D If AB

22

11

then , of midpoint the is E AC

and ,|| BCDE BCDE2

1

5

5

8

8

x x 22

2

1

x 11

Use theorem 9 – 7 to find the length of segment DE.


A, B, and C are midpoints of the sides of ΔMNP.

Complete each statement.

C

BA

N P

M

2) If BC = 14, then MN = ____

1) MP || ____AC

28

3) If mMNP = s, then mBCP = ___s

4) If MP = 18x, then AC = __9x


A, B, and C are midpoints of the sides of ΔDEF.

2) Find the perimeter of ΔABC

1) Find DE, EF, and FD.14; 10; 16

20

4) Find the ratio of the perimeter of ΔABC to the perimeter of ΔDEF.

20:40 =

C

BA

D F

E

8

753) Find the perimeter of ΔDEF 40

1:2


A

D

C

B

ABCD is a quadrilateral.

E is the midpoint of AD

F is the midpoint of DC

H is the midpoint of CB

G is the midpoint of BA

Q1) What can you say about EF and GH ?

E

G

H

F

(Hint: Draw diagonal AC .)

They are parallel

Q2) What kind of figure is EFHG ? Parallelogram


Proportional Parts and Parallel Lines Proportional Parts and Parallel Lines

You will learn to identify and use the relationships betweenparallel lines and proportional parts.

Nothing New!


On your given paper, draw two (transversals) lines intersecting the parallel lines.

C

D

E

F

A

BLabel the intersections of the transversals and the parallel lines,as shown here.

Measure AB, BC, DE, and EF.

,Calculate each set of ratios:BC

AB

EF

DE

AC

AB

DF

DE,

Do the parallel lines divide the transversals proportionally? Yes


Theorem 9-8

If three or more parallel lines intersect two transversals,the lines divide the transversals proportionally.

l

m

n

B

A

C F

E

D

If l || m || n

AC

BC

DF

EF=Then

BC

AB

EF

DE= ,

AC

AB

DF

DE= and,


a

b

c

H

G

J W

V

U

18

12

x

15

Find the value of x.

HJ

GH

VW

UV=

18

12

x

15=

12x = 18(15)

12x = 270

x = 222

1


Theorem 9-9

If three or more parallel lines cut off congruent segments onone transversal, then they cut off congruent segments onevery transversal.

l

m

n

B

A

C F

E

D

If l || m || n and

Then

AB BC,

DE EF.


(x +3)

10

(2x – 2)

10

Find the value of x.

F

(x + 3) = (2x – 2)

x + 3 = 2x – 2

5 = x

Theorem 9 - 9

8 8

ED

C

B

A

DE EF

Since AB BC,


Perimeters and Similarity Perimeters and Similarity

You will learn to identify and use proportional relationships ofsimilar triangles.

1) Scale Factor


These right triangles are similar! Therefore, the measures of their corresponding sides are ___________.

Is there a relationship between the measures of the perimeters of the twotriangles?

8

6 10

12

159

proportional

We know that6

9

8

12

10

15= =

Use the ____________ theoremto calculate the length of thehypotenuse.

Pythagorean

222 bac

2

3=

perimeter of small Δ

perimeter of large Δ=

9 + 12 + 15

6 + 8 + 10=

36

24=

32


perimeter of ΔABCperimeter of ΔDEF

=DEAB

=

Theorem

9-10

If two triangles are similar, then

A

C B

F E

D

the measures of the corresponding perimeters are proportional to the measures of the corresponding sides.

If ΔABC ~ ΔDEF, then

EFBC

=FDCA


27 = 13.5x

The perimeter of ΔRST is 9 units, and ΔRST ~ ΔMNP.

Find the value of each variable. M

N

4.5

P

R

S

T 3

6

z

Yx

perimeter of ΔMNP

perimeter of ΔRST

RSMN

=

13.59x

3=

Theorem 9-10

The perimeter of ΔMNP is 3 + 6 + 4.5

Cross Products

x = 2

RSMN

=STNP

RSMN

=TRPM

23

= y6

23

= z4.5

3y = 12 3z = 9

y = 4 z = 3


DEAB

=EFBC

FDCA

=

Each ratio is equivalent to 21

If ΔABC ~ ΔDEF, then

The ratio found by comparing the measures of corresponding sides of similar triangles is called the constant of proportionality or the ___________.scale factor

A

B C7

53

D

E F14

106

or63

=147

=105

The scale factor of ΔABC to ΔDEF is21

The scale factor of ΔDEF to ΔABC is12



A

E

D

C

B

I

H

G

F

Step 1) On a piece of lined paper, pick a point on one of the lines and label it A.

Use a straightedge and protractor to draw A so that mA < 90 and only the vertex lies on the line.

Step 2) Extend one side of A down four lines. Label this point E.

Do the same for the other side of A. Label this point I.Now connect points E and I to form ΔAEI.

Step 3) Label the points where the horizontal lines intersect segment AG (B through D).

Label the points where the horizontal lines intersect segment AI (F through H).

AIAHAGAEADAC and , , , , , Measure

GI

AG

CE

AC and ,

:ratios following the compare and Calculate

AI

AH

AE

AD and ,

What can you conclude about the lines through the sides of ΔAEI andparallel to segment EI?

This activity suggests Theorem 9-5.

Documents

§ 9.1 Using Ratios and Proportions § 9.4 Proportional Parts and Triangles § 9.3 Similar Triangles § 9.2 Similar Polygons § 9.6 Proportional Parts