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Chapter 6Resource Masters

Geometry

Reading to Learn MathematicsVocabulary Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

66

© Glencoe/McGraw-Hill vii Glencoe Geometry

Voca

bula

ry B

uild

erThis is an alphabetical list of the key vocabulary terms you will learn in Chapter 6.As you study the chapter, complete each term’s definition or description. Rememberto add the page number where you found the term. Add these pages to yourGeometry Study Notebook to review vocabulary at the end of the chapter.

Vocabulary Term Found on Page Definition/Description/Example

cross products

extremes

fractal

iteration

ID·uh·RAY·shuhn

means

(continued on the next page)

⎧ ⎪ ⎨ ⎪ ⎩

© Glencoe/McGraw-Hill viii Glencoe Geometry

Vocabulary Term Found on Page Definition/Description/Example

midsegment

proportion

ratio

scale factor

self-similar

similar polygons

Reading to Learn MathematicsVocabulary Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

66

Learning to Read MathematicsProof Builder

NAME ______________________________________________ DATE ____________ PERIOD _____

66

© Glencoe/McGraw-Hill ix Glencoe Geometry

Proo

f Bu

ilderThis is a list of key theorems and postulates you will learn in Chapter 6. As you

study the chapter, write each theorem or postulate in your own words. Includeillustrations as appropriate. Remember to include the page number where youfound the theorem or postulate. Add this page to your Geometry Study Notebookso you can review the theorems and postulates at the end of the chapter.

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 6.1Side-Side-Side (SSS) Similarity

Theorem 6.2Side-Angle-Side (SAS) Similarity

Theorem 6.3

Theorem 6.4Triangle Proportionality Theorem

Theorem 6.5Converse of the TriangleProportionality Theorem

Theorem 6.6Triangle Midsegment Theorem

(continued on the next page)

© Glencoe/McGraw-Hill x Glencoe Geometry

Theorem or Postulate Found on Page Description/Illustration/Abbreviation

Theorem 6.7Proportional Perimeters Theorem

Theorem 6.8

Theorem 6.9

Theorem 6.10

Theorem 6.11Angle Bisector Theorem

Postulate 6.1Angle-Angle (AA) Similarity

Learning to Read MathematicsProof Builder (continued)

NAME ______________________________________________ DATE ____________ PERIOD _____

66

Study Guide and InterventionProportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

© Glencoe/McGraw-Hill 295 Glencoe Geometry

Less

on

6-1

Write Ratios A ratio is a comparison of two quantities. The ratio a to b, where b is not zero, can be written as !

ab! or a:b. The ratio of two quantities is sometimes called a scale

factor. For a scale factor, the units for each quantity are the same.

In 2002, the Chicago Cubs baseball team won 67 games out of 162.Write a ratio for the number of games won to the total number of games played.To find the ratio, divide the number of games won by the total number of games played. The result is !1

6672!, which is about 0.41. The Chicago Cubs won about 41% of their games in 2002.

A doll house that is 15 inches tall is a scale model of a real housewith a height of 20 feet. What is the ratio of the height of the doll house to theheight of the real house?To start, convert the height of the real house to inches.20 feet " 12 inches per foot # 240 inchesTo find the ratio or scale factor of the heights, divide the height of the doll house by theheight of the real house. The ratio is 15 inches:240 inches or 1:16. The height of the doll house is !1

16! the height of the real house.

1. In the 2002 Major League baseball season, Sammy Sosa hit 49 home runs and was atbat 556 times. Find the ratio of home runs to the number of times he was at bat.

2. There are 182 girls in the sophomore class of 305 students. Find the ratio of girls to total students.

3. The length of a rectangle is 8 inches and its width is 5 inches. Find the ratio of length to width.

4. The sides of a triangle are 3 inches, 4 inches, and 5 inches. Find the scale factor betweenthe longest and the shortest sides.

5. The length of a model train is 18 inches. It is a scale model of a train that is 48 feet long.Find the scale factor.

Example 1Example 1

Example 2Example 2

ExercisesExercises


Use Properties of Proportions A statement that two ratios are equal is called aproportion. In the proportion !

ab! # !d

c!, where b and d are not zero, the values a and d are

the extremes and the values b and c are the means. In a proportion, the product of themeans is equal to the product of the extremes, so ad # bc.

!ab! # !d

c!

a $ d # b $ c↑ ↑

extremes means

Solve !196! " !

2x7!.

!196! # !

2x7!

9 $ x # 16 $ 27 Cross products

9x # 432 Multiply.

x # 48 Divide each side by 9.

A room is 49 centimeters by 28 centimeters on a scale drawing of ahouse. For the actual room, the larger dimension is 14 feet. Find the shorterdimension of the actual room.If x is the room’s shorter dimension, then

!2489! # !1

x4!

49x # 392 Cross products

x # 8 Divide each side by 49.

The shorter side of the room is 8 feet.

Solve each proportion.

1. !12! # !

2x8! 2. !

38! # !2

y4! 3. !

xx

%%

222

! # !3100!

4. !183.2! # !

9y! 5. !

2x8% 3! # !

54! 6. !

xx

%&

11! # !

34!

Use a proportion to solve each problem.

7. If 3 cassettes cost $44.85, find the cost of one cassette.

8. The ratio of the sides of a triangle are 8:15:17. If the perimeter of the triangle is 480 inches, find the length of each side of the triangle.

9. The scale on a map indicates that one inch equals 4 miles. If two towns are 3.5 inchesapart on the map, what is the actual distance between the towns?

shorter dimension!!!longer dimension

Study Guide and Intervention (continued)

Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

Example 1Example 1

Example 2Example 2

ExercisesExercises

Skills PracticeProportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1


Less

on

6-1

1. FOOTBALL A tight end scored 6 touchdowns in 14 games. Find the ratio of touchdownsper game.

2. EDUCATION In a schedule of 6 classes, Marta has 2 elective classes. What is the ratioof elective to non-elective classes in Marta’s schedule?

3. BIOLOGY Out of 274 listed species of birds in the United States, 78 species made theendangered list. Find the ratio of endangered species of birds to listed species in theUnited States.

4. ART An artist in Portland, Oregon, makes bronze sculptures of dogs. The ratio of theheight of a sculpture to the actual height of the dog is 2:3. If the height of the sculptureis 14 inches, find the height of the dog.

5. SCHOOL The ratio of male students to female students in the drama club at CampbellHigh School is 3:4. If the number of male students in the club is 18, what is the numberof female students?


6. !25! # !4

x0! 7. !1

70! # !

2x1! 8. !

250! # !

46x!

9. !54x! # !

385! 10. !

x %3

1! # !

72! 11. !

135! # !

x &5

3!

Find the measures of the sides of each triangle.

12. The ratio of the measures of the sides of a triangle is 3:5:7, and its perimeter is 450 centimeters.

13. The ratio of the measures of the sides of a triangle is 5:6:9, and its perimeter is 220 meters.

14. The ratio of the measures of the sides of a triangle is 4:6:8, and its perimeter is 126 feet.

15. The ratio of the measures of the sides of a triangle is 5:7:8, and its perimeter is 40 inches.


1. NUTRITION One ounce of cheddar cheese contains 9 grams of fat. Six of the grams of fatare saturated fats. Find the ratio of saturated fats to total fat in an ounce of cheese.

2. FARMING The ratio of goats to sheep at a university research farm is 4:7. The numberof sheep at the farm is 28. What is the number of goats?

3. ART Edward Hopper’s oil on canvas painting Nighthawks has a length of 60 inches anda width of 30 inches. A print of the original has a length of 2.5 inches. What is the widthof the print?


4. !58! # !1

x2! 5. !1.

x12! # !

15! 6. !2

67x! # !

43!

7. !x %

32

! # !89! 8. !

3x4& 5! # !

&75! 9. !

x &4

2! # !

x %2

4!

Find the measures of the sides of each triangle.

10. The ratio of the measures of the sides of a triangle is 3:4:6, and its perimeter is 104 feet.

11. The ratio of the measures of the sides of a triangle is 7:9:12, and its perimeter is 84 inches.

12. The ratio of the measures of the sides of a triangle is 6:7:9, and its perimeter is 77 centimeters.

Find the measures of the angles in each triangle.

13. The ratio of the measures of the angles is 4:5:6.

14. The ratio of the measures of the angles is 5:7:8.

15. BRIDGES The span of the Benjamin Franklin suspension bridge in Philadelphia,Pennsylvania, is 1750 feet. A model of the bridge has a span of 42 inches. What is theratio of the span of the model to the span of the actual Benjamin Franklin Bridge?

Practice Proportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

Reading to Learn MathematicsProportions

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1


Less

on

6-1

Pre-Activity How do artists use ratios?

Read the introduction to Lesson 6-1 at the top of page 282 in your textbook.

Estimate the ratio of length to width for the background rectangles inTiffany’s Clematis Skylight.

Reading the Lesson

1. Match each description in the first column with a word or phrase from the secondcolumn.

a. The ratio of two corresponding quantities i. proportion

b. r and u in the equation !rs! # !u

t! ii. cross products

c. a comparison of two quantities iii. means

d. ru and st in the equation !rs! # !u

t! iv. scale factor

e. an equation stating that two ratios are equal v. extremes

f. s and t in the equation !rs! # !u

t! vi. ratio

2. If m, n, p, and q are nonzero numbers such that !mn! # !

pq!, which of the following

statements could be false?

A. np # mq B. !np

! # !mq!

C. mp # nq D. qm # pn

E. !mn! # !p

q! F. !p

q! # !m

n!

G. m:p # n:q H. m:n # p:q

3. Write two proportions that match each description.

a. Means are 5 and 8; extremes are 4 and 10.

b. Means are 5 and 4; extremes are positive integers that are different from means.

Helping You Remember

4. Sometimes it is easier to remember a mathematical idea if you put it into words withoutusing any mathematical symbols. How can you use this approach to remember theconcept of equality of cross products?


Golden Rectangles

Use a straightedge, compass, and the instructions belowto construct a golden rectangle.

1. Construct square ABCD with sides of 2 cm.

2. Construct the midpoint of A!B!. Call the midpoint M.

3. Draw AB!"#. Set your compass at an opening equal to MC.Use M as the center to draw an arc that intersects AB!"#. Call the point of intersection P.

4. Construct a line through P that is perpendicular to AB!"#.

5. Draw DC!"# so that it intersects the perpendicular line in step 4.Call the intersection point Q. APQD is a golden rectanglebecause the ratio of its length to its width is 1.618. Check this conclusion by finding the value of !QAP

P!. Rectangles whose sides

have this ratio are, it is said, the most pleasing to the human eye.

A figure consisting of similar golden rectangles is shown below. Use a compassand the instructions below to draw quarter-circle arcs that form a spiral like thatfound in the shell of a chambered nautilus.

6. Using A as a center, draw an arc that passes through B and C.

7. Using D as a center, draw an arc that passes through C and E.

8. Using F as a center, draw an arc that passes through E and G.

9. Using H as a center, draw an arc that passes through G and J.

10. Using K as a center, draw an arc that passes through J and L.

11. Using M as a center, draw an arc that passes through L and N.

M NH

L FD

JK

B A

C

G

E

A M B P

D C Q

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-16-1

Study Guide and InterventionSimilar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on

6-2

Identify Similar Figures

Determine whether the triangles are similar.Two polygons are similar if and only if their corresponding angles are congruent and their corresponding sides are proportional.

!C " !Z because they are right angles, and !B " !X.By the Third Angle Theorem, !A " !Y.

For the sides, !BX

CZ! # !

2203!, !

BX

AY! # # !

2203!, and !

AY

CZ! # !

2203!.

The side lengths are proportional. So "BCA # "XZY.

Is polygon WXYZ ! polygon PQRS?

For the sides, !WPQ

X! # !

182! # !

32!, !Q

XYR! # !

1182! # !

32!, !R

YZS! # !

1150! # !

32!,

and !ZSWP! # !

96! # !

32!. So corresponding sides are proportional.

Also, !W " !P, !X " !Q, !Y " !R, and !Z " !S, so corresponding angles are congruent. We can conclude that polygon WXYZ # polygon PQRS.

Determine whether each pair of figures is similar. If they are similar, give theratio of corresponding sides.

1. 2.

3. 4. 10y

8y

37#

37#116#27#

5x

4x15z

12z

22

11

1610

equilateral triangles

15

9

12

18

XW

Z Y 10

6

8

12

QP

S R

20$2!!23$2!

20 23

2320 20$%2 23$%245#

45#C A X Z

B Y

ExercisesExercises

Example 1Example 1

Example 2Example 2


Scale Factors When two polygons are similar, the ratio of the lengths of corresponding sides is called the scale factor. At the right, "ABC # "XYZ. The scale factor of "ABC to "XYZ is 2 and the scale factor of "XYZ to "ABC is !

12!.

3 cm

6 cm

8 cm

10 cm5 cm4 cm

Z

Y

XC

A

B


Similar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

The two polygons aresimilar. Find x and y.

Use the congruent angles to write thecorresponding vertices in order."RST # "MNPWrite proportions to find x and y.

!3126! # !1

x3! !

3y8! # !

3126!

16x # 32(13) 32y # 38(16)x # 26 y # 19

x

y38

3216 13

T

S

R P

N

M

!ABC ! !CDE. Find the scale factor and find the lengths of C"D"and D"E".

AC # 3 & 0 # 3 and CE # 9 & 3 # 6. Thescale factor of "CDE to "ABC is 6:3 or 2:1.Using the Distance Formula,AB # $1 % 9! # $10! and BC # $4 % 9! # $13!. The lengths of thesides of "CDE are twice those of "ABC,so DC # 2(BA) or 2$10! and DE # 2(BC) or 2$13!.

x

y

A(0, 0)

B(1, 3)

C(3, 0)

E(9, 0)

D

Example 1Example 1 Example 2Example 2

ExercisesExercises

Each pair of polygons is similar. Find x and y.

1. 2.

3. 4.

5. In Example 2 above, point D has coordinates (5, 6). Use the Distance Formula to verifythe lengths of C!D! and D!E!.

40

36x

30

18 y12

y $ 1

18

24

x18

10

y

4.5x

2.5

5

9

12

y8

10x

Skills PracticeSimilar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on

6-2

Determine whether each pair of figures is similar. Justify your answer.

1. 2.

Each pair of polygons is similar. Write a similarity statement, and find x, themeasure(s) of the indicated side(s), and the scale factor.

3. G!H! 4. S!T! and S!U!

5. W!T! 6. T!S! and S!P!

10

8

x $ 2

x % 1

SP

M N

L

T

9

103

x $ 1

5

P

S

RU V

WT

Q

x $ 5

x $ 5

9

3

44 S

W

XY

T

U26

14

12 13

B C

DA

13

76 x

G

E H

F

7.5

7.5

7.57.5

Z Y

XW

3

3

33S R

QP

10.5

6 9

59# 35#A C

B

7

4 659# 35#D F

E


Determine whether each pair of figures is similar. Justify your answer.

1. 2.

Each pair of polygons is similar. Write a similarity statement, and find x, themeasure(s) of the indicated side(s), and the scale factor.

3. L!M! and M!N! 4. D!E! and D!F!

5. COORDINATE GEOMETRY Triangle ABC has vertices A(0, 0), B(&4, 0), and C(&2, 4).The coordinates of each vertex are multiplied by 3 to create "AEF. Show that "AEF issimilar to "ABC.

6. INTERIOR DESIGN Graham used the scale drawing of his living room to decide where to place furniture. Find the dimensions of theliving room if the scale in the drawing is 1 inch # 4.5 feet.

4 in.

21–2 in.

6 12

40#

40#

x $ 1

x % 3AF

E

D

B C14

10

x $ 6

x $ 9

C N MD

A B LP

2412

14 16 2118

VUAC

B T

25 12

9

14.4

1524

20

15

JM Q R

SP

KL

Practice Similar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

Reading to Learn MathematicsSimilar Polygons

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2


Less

on

6-2

Pre-Activity How do artists use geometric patterns?

Read the introduction to Lesson 6-2 at the top of page 289 in your textbook.• Describe the figures that have similar shapes.

• What happens to the figures as your eyes move from the center to theouter edge?

Reading the Lesson1. Complete each sentence.

a. Two polygons that have exactly the same shape, but not necessarily the same size,are .

b. Two polygons are congruent if they have exactly the same shape and the same .

c. Two polygons are similar if their corresponding angles are and their corresponding sides are .

d. Two polygons are congruent if their corresponding angles are and their corresponding sides are .

e. The ratio of the lengths of corresponding sides of two similar figures is called the .

f. Multiplying the coordinates of all points of a figure in the coordinate plane by a scale factor to get a similar figure is called a .

g. If two polygons are similar with a scale factor of 1, then the polygons are .

2. Determine whether each statement is always, sometimes, or never true.a. Two similar triangles are congruent.b. Two equilateral triangles are congruent.c. An equilateral triangle is similar to a scalene triangle.d. Two rectangles are similar.e. Two isosceles right triangles are congruent.f. Two isosceles right triangles are similar.g. A square is similar to an equilateral triangle.h. Two acute triangles are similar.i. Two rectangles in which the length is twice the width are similar.j. Two congruent polygons are similar.

Helping You Remember3. A good way to remember a new mathematical vocabulary term is to relate it to words

used in everyday life. The word scale has many meanings in English. Give three phrasesthat include the word scale in a way that is related to proportions.


Constructing Similar PolygonsHere are four steps for constructing a polygon that is similar to and withsides twice as long as those of an existing polygon.

Step 1 Choose any point either inside or outside the polygon and label it O.Step 2 Draw rays from O through each vertex of the polygon.Step 3 For vertex V, set the compass to length OV. Then locate a new point

V' on ray OV such that VV' # OV. Thus, OV' # 2(OV ).Step 4 Repeat Step 3 for each vertex. Connect points V', W', X' and Y' to

form the new polygon.

Two constructions of polygons similar to and with sides twice those of VWXYare shown below. Notice that the placement of point O does not affect the sizeor shape of V'W'X'Y', only its location.

Trace each polygon. Then construct a similar polygon with sides twice as long asthose of the given polygon.

1. 2. D E F

GC

A

B

VW

Y X

VW

Y X

V'

W'

X'Y'

O

VW

YX

V'

W'

X'Y'

O

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-26-2

3. Explain how to construct a similarpolygon with sides three times the lengthof those of polygon HIJKL. Then do theconstruction.

4. Explain how to construct a similar polygon 1!

12! times the length of those of

polygon MNPQRS. Then do theconstruction.

M

S

R

N

Q

P

H I

LK

J

Study Guide and InterventionSimilar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on

6-3

Identify Similar Triangles Here are three ways to show that two triangles are similar.

AA Similarity Two angles of one triangle are congruent to two angles of another triangle.

SSS Similarity The measures of the corresponding sides of two triangles are proportional.

SAS SimilarityThe measures of two sides of one triangle are proportional to the measures of two corresponding sides of another triangle, and the included angles are congruent.

Determine whether thetriangles are similar.

!DAC

F! # !69! # !

23!

!BE

CF! # !1

82! # !

23!

!DAB

E! # !1105! # !

23!

"ABC # "DEF by SSS Similarity.

15

129

D E

F

10

86

A B

C

Determine whether thetriangles are similar.

!34! # !

68!, so !MQR

N! # !

NRS

P!.

m!N # m!R, so !N " !R."NMP # "RQS by SAS Similarity.

8

4

70#

Q

R S6

370#

M

N P


ExercisesExercises

Determine whether each pair of triangles is similar. Justify your answer.

1. 2.

3. 4.

5. 6. 3220

154025

24

3926

1624

18

8

65#9

465#

36

2018

24


Use Similar Triangles Similar triangles can be used to find measurements.


Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

!ABC ! !DEF.Find x and y.

!DAC

F! # !BE

CF! !D

ABE! # !

BE

CF!

# !198! !1

y8! # !

198!

18x # 9(18$3!) 9y # 324x # 9$3! y # 36

18$3!!x

18$%3

B

CA

18 18 9y

x

E

FD

A person 6 feet tall castsa 1.5-foot-long shadow at the same timethat a flagpole casts a 7-foot-longshadow. How tall is the flagpole?

The sun’s rays form similar triangles.Using x for the height of the pole, !

6x! # !

17.5!,

so 1.5x # 42 and x # 28.The flagpole is 28 feet tall.

7 ft

6 ft?

1.5 ft


ExercisesExercises

Each pair of triangles is similar. Find x and y.

1. 2.

3. 4.

5. 6.

7. The heights of two vertical posts are 2 meters and 0.45 meter. When the shorter postcasts a shadow that is 0.85 meter long, what is the length of the longer post’s shadow to the nearest hundredth?

y

x32

10

2230yx

9

8

7.2

16

yx

6038.6

23

3024

36$%2yx

3636

20

13

26y

x $ 235

13

10 20y

x

Skills PracticeSimilar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on

6-3


1. 2.

3. 4.

ALGEBRA Identify the similar triangles, and find x and the measures of theindicated sides.

5. A!C! and E!D! 6. J!L! and L!M!

7. E!H! and E!F! 8. U!T! and R!T!

6

14

S

U

V

R T

x % 4

x % 612

9

6

9

D

E

FH

Gx $ 5

16

4M

NL

J

K

x $ 18

x % 3

1512

D

E CB

Ax $ 1

x $ 5

30#60#RQ

M

S

K

T

2170#

15

MJ

P

1470#10

U

S

T

12

128

9

9 6

C P

QR

A

B

2014

13 12219

S W X

Y

R

T



1. 2.

ALGEBRA Identify the similar triangles, and find x and the measures of theindicated sides.

3. L!M! and Q!P! 4. N!L! and M!L!

Use the given information to find each measure.

5. If T!S! || Q!R!, TS # 6, PS # x % 7, 6. If E!F! || H!I!, EF # 3, EG # x % 1,QR # 8, and SR # x & 1, HI # 4, and HG # x % 3,find PS and PR. find EG and HG.

INDIRECT MEASUREMENT For Exercises 7 and 8, use the following information.A lighthouse casts a 128-foot shadow. A nearby lamppost that measures 5 feet 3 inches castsan 8-foot shadow.

7. Write a proportion that can be used to determine the height of the lighthouse.

8. What is the height of the lighthouse?

F

GE

H

I

S RP

QT

8

N

J LK

Mx $ 5

6x $ 2

24

12

18N

P

QL

M

x $ 3x % 1

18

16 12.516

14

11

M

L N

S R

T42#

42#

24

1812

16J

A K

Y S

W

Practice Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

Reading to Learn MathematicsSimilar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3


Less

on

6-3

Pre-Activity How do engineers use geometry?

Read the introduction to Lesson 6-3 at the top of page 298 in your textbook.• What does it mean to say that triangular shapes result in rigid

construction?

• What would happen if the shapes used in the construction werequadrilaterals?

Reading the Lesson1. State whether each condition guarantees that two triangles are congruent or similar.

If the condition guarantees that the triangles are both similar and congruent, writecongruent. If there is not enough information to guarantee that the triangles will becongruent or similar, write neither.a. Two sides and the included angle of one triangle are congruent to two sides and the

included angle of the other triangle.b. The measures of all three pairs of corresponding sides are proportional.c. Two angles of one triangle are congruent to two angles of the other triangle.d. Two angles and a nonincluded side of one triangle are congruent to two angles and

the corresponding nonincluded side of the other triangle.e. The measures of two sides of a triangle are proportional to the measures of two

corresponding sides of another triangle, and the included angles are congruent.

f. The three sides of one triangle are congruent to the three sides of the other triangle.

g. The three angles of one triangle are congruent to the three angles of the other triangle.

h. One acute angle of a right triangle is congruent to one acute angle of another righttriangle.

i. The measures of two sides of a triangle are proportional to the measures of two sidesof another triangle.

2. Identify each of the following as an example of a reflexive, symmetric, or transitive property.a. If "RST # "UVW, then "UVW # "RST.b. If "RST # "UVW and "UVW # "OPQ, then "RST # "OPQ.c. "RST # "RST

Helping You Remember3. A good way to remember something is to explain it to someone else. Suppose one of your

classmates is having trouble understanding the difference between SAS for congruenttriangles and SAS for similar triangles. How can you explain the difference to him?


Ratio Puzzles with TrianglesIf you know the perimeter of a triangle and the ratios of the sides, you canfind the lengths of the sides.

The perimeter of a triangle is 84 units. The sides havelengths r, s, and t. The ratio of s to r is 5:3, and the ratio of t to r is2:1. Find the length of each side.

Since both ratios contain r, rewrite one or both ratios to make r the same. Youcan write the ratio of t to r as 6:3. Now you can write a three-part ratio.r : s : t # 3: 5: 6

There is a number x such that r # 3x, s # 5x, and t # 6x. Since you know theperimeter, 84, you can use algebra to find the lengths of the sides.

r % s % t # 843x % 5x % 6x # 84

14x # 84x # 6

3x # 18, 5x # 30, 6x # 36

So r # 18, s # 30, and t # 36.

Find the lengths of the sides of each triangle.

1. The perimeter of a triangle is 75 units. The sides have lengths a, b, and c. The ratio of b to a is 3:5, and the ratio of c to a is 7:5. Find the length of each side.

2. The perimeter of a triangle is 88 units. The sides have lengths d, e, and f. The ratio of e to d is 3:1, and the ratio of f to e is 10:9. Find the length of each side.

3. The perimeter of a triangle is 91 units. The sides have lengths p, q, and r. The ratio of p to r is 3:1, and the ratio of q to r is 5:2. Find the length of each side.

4. The perimeter of a triangle is 68 units. The sides have lengths g, h, and j. The ratio of j to g is 2:1, and the ratio of h to g is 5:4. Find the length of each side.

5. Write a problem similar to those above involving ratios in triangles.

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-36-3

ExampleExample

Study Guide and InterventionParallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

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Proportional Parts of Triangles In any triangle, a line parallel to one side of a triangle separates the other two sides proportionally. The converse is also true.

If X and Y are the midpoints of R!T! and S!T!, then X!Y! is amidsegment of the triangle. The Triangle Midsegment Theoremstates that a midsegment is parallel to the third side and is half its length.

If XY!"# || RS!"#, then !RXTX! # !Y

SYT!.

If !RXTX! # !Y

SYT!, then XY!"# || RS!"#.

If X!Y! is a midsegment, then XY!"# || RS!"# and XY # !12!RS.

Y ST

R

X

In !ABC, E"F" || C"B". Find x.

Since E!F! || C!B!, !FA

BF! # !E

AEC!.

!xx

%%

222

! # !168!

6x % 132 # 18x % 3696 # 12x8 # x

F

C

BA

18

x $ 22 x $ 2

6E

A triangle has verticesD(3, 6), E(%3, %2), and F(7, %2).Midsegment G"H" is parallel to E"F". Find the length of G"H".G!H! is a midsegment, so its length is one-half that of E!F!. Points E and F have thesame y-coordinate, so EF # 7 & (&3) # 10.The length of midsegment G!H! is 5.


ExercisesExercises

Find x.

1. 2. 3.

4. 5. 6.

7. In Example 2, find the slope of E!F! and show that E!F! || G!H!.

30 10x $ 10

x11x $ 12

x33

30 10

24x

35

x9

20

x

185

5 x

7


Divide Segments Proportionally When three or more parallel lines cut two transversals,they separate the transversals into proportional parts. If the ratio of the parts is 1, then the parallellines separate the transversals into congruent parts.

If !1 || !2 || !3, If !4 || !5 || !6 and

then !ab! # !d

c!. !

uv! # 1, then !

wx! # 1.

Refer to lines !1, !2 ,and !3 above. If a " 3, b " 8, and c " 5, find d.

!1 || !2 || !3 so !38! # !d

5!. Then 3d # 40 and d # 13!

13!.

Find x and y.

1. 2.

3. 4.

5. 6.16 32

y $ 3y

3 4

x $ 4x

5

8 y $ 2

y2x $ 4

3x % 1 2y $ 2

3y

2x % 6

x $ 3

12

x $ 12 12

3x5x

a

bd

u v

xw

ctn

m

s !1

!4 !5 !6

!2

!3


Parallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

ExampleExample

ExercisesExercises

Skills PracticeParallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

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1. If JK # 7, KH # 21, and JL # 6, 2. Find x and TV if RU # 8, US # 14,find LI. TV # x & 1 and VS # 17.5.

Determine whether B!C! || D!E!.

3. AD # 15, DB # 12, AE # 10, and EC # 8

4. BD # 9, BA # 27, and CE is one third of EA

5. AE # 30, AC # 45, and AD is twice DB

COORDINATE GEOMETRY For Exercises 6–8, use the following information.Triangle ABC has vertices A(&5, 2), B(1, 8), and C(4, 2). Point Dis the midpoint of A!B! and E is the midpoint of A!C!.

6. Identify the coordinates of D and E.

7. Show that B!C! is parallel to D!E!.

8. Show that DE # !12!BC.

9. Find x and y. 10. Find x and y.

3–2x $ 2

x $ 3

2y % 1

3y % 52x $ 1 3y % 8

y $ 5x $ 7

x

y

O

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B

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E

C

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D

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1. If AD # 24, DB # 27, and EB # 18, 2. Find x, QT, and TR if QT # x % 6,find CE. SR # 12, PS # 27, and TR # x & 4.

Determine whether J"K" || N"M".

3. JN # 18, JL # 30, KM # 21, and ML # 35

4. KM # 24, KL # 44, and NL # !56!JN

COORDINATE GEOMETRY For Exercises 5 and 6, use the following information.Triangle EFG has vertices E(&4, &1), F(2, 5), and G(2, &1). Point Kis the midpoint of E!G! and H is the midpoint of F!G!.

5. Show that E!F! is parallel to K!H!.

6. Show that KH # !12!EF.

7. Find x and y. 8. Find x and y.

9. MAPS The distance from Wilmington to Ash Grove along Kendall is 820 feet and along Magnolia, 660 feet. If the distance between Beech and Ash Grove along Magnolia is 280 feet, what is the distance between the two streets alongKendall?

Ash Grove

Beech

Magnolia Kendall

Wilmington

4–3y $ 2

3x % 4 x $ 1

4y % 43x % 4 2–

3y $ 31–3y $ 65–

4x $ 3

x

y

O

E G

F

H

K

N LJ

KM

S

Q

TRP

D

E

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BA

Practice Parallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

Reading to Learn MathematicsParallel Lines and Proportional Parts

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4


Less

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6-4

Pre-Activity How do city planners use geometry?


Use a geometric idea to explain why the distance between Chicago Avenueand Ontario Street is shorter along Michigan Avenue than along LakeShore Drive.

Reading the Lesson

1. Provide the missing words to complete the statement of each theorem. Then state thename of the theorem.

a. If a line intersects two sides of a triangle and separates the sides into corresponding

segments of lengths, then the line is to the

third side.

b. A midsegment of a triangle is to one side of the triangle and its

length is the length of that side.

c. If a line is to one side of a triangle and intersects the other two sides

in distinct points, then it separates these sides into

of proportional length.

2. Refer to the figure at the right.

a. Name the three midsegments of "RST.

b. If RS # 8, RU # 3, and TW # 5, find the length of each of themidsegments.

c. What is the perimeter of "RST?

d. What is the perimeter of "UVW?

e. What are the perimeters of "RUV, "SVW, and "TUW?

f. How are the perimeters of each of the four small triangles related to the perimeter ofthe large triangle?

g. Would the relationship that you found in part f apply to any triangle in which themidpoints of the three sides are connected?


3. A good way to remember a new mathematical term is to relate it to other mathematicalvocabulary that you already know. What is an easy way to remember the definition ofmidsegment using other geometric terms?

R

U

V

W

T

S


Parallel Lines and Congruent PartsThere is a theorem stating that if three parallel lines cut off congruent segments on onetransversal, then they cut off congruent segments on any transversal. This can be shown forany number of parallel lines. The following drafting technique uses this fact to divide asegment into congruent parts.

A!B! is to be separated into five congruent parts. This can be done very accurately without using a ruler. All that is needed is a compass and a piece of notebook paper.

Step 1 Hold the corner of a piece of notebook paper at point A.

Step 2 From point A, draw a segment along the paper that is five spaces long. Mark where the lines of the notebook paper meet the segment. Label the fifth point, P.

Step 3 Draw P!B!. Through each of the other marks on A!P!, construct a line parallel to B!P!. The points where these lines intersect A!B! will divide A!B! into five congruent segments.

Use a compass and a piece of notebook paper to divide each segment into the given number of congruent parts.

1. six congruent parts

2. seven congruent parts A B

A B

A

P

B

A B

P

A B

A B

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-46-4

Study Guide and InterventionParts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


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Perimeters If two triangles are similar, their perimeters have the same proportion as the corresponding sides.

If "ABC # "RST, then

!AR

BS

%%

BST

C%%

RA

TC

! # !AR

BS! # !

BST

C! # !R

ACT!.

Use the diagram above with !ABC ! !RST. If AB " 24 and RS " 15,find the ratio of their perimeters.Since "ABC # "RST, the ratio of the perimeters of "ABC and "RST is the same as theratio of corresponding sides.

Therefore # !2145!

# !85!

Each pair of triangles is similar. Find the perimeter of the indicated triangle.

1. "XYZ 2. "BDE

3. "ABC 4. "XYZ

5. "ABC 6. "RST

25

20

10

P T

SM

R

95

13

12

A D

E

C

B

5 68.7

13

10

X Y

Z T

R S18

20

25 15 17

8

NB

A C PM

36

9

11C

DE

A

B44

24

20 22 10

TRX Z

YS

perimeter of "ABC!!!perimeter of "RST

A C

BS

R T

ExampleExample

ExercisesExercises


Special Segments of Similar Triangles When two triangles are similar,corresponding altitudes, angle bisectors, and medians are proportional to the correspondingsides. Also, in any triangle an angle bisector separates the opposite side into segments thathave the same ratio as the other two sides of the triangle.


Parts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

In the figure,!ABC ! !XYZ, with angle bisectors asshown. Find x.

Since "ABC # "XYZ, the measures of theangle bisectors are proportional to themeasures of a pair of corresponding sides.

!AX

BY! # !Y

BWD!

!2x4! # !

180!

10x # 24(8)10x # 192

x # 19.2

2410

DW

Y

ZXC

B

A

8x

S"U" bisects "RST. Find x.

Since S!U! is an angle bisector, !RTU

U! # !

RTS

S!.

!2x0! # !

1350!

30x # 20(15)30x # 300

x # 10

x 20

3015

UT

S

R


ExercisesExercises

Find x for each pair of similar triangles.

1. 2.

3. 4.

5. 6. 2x % 2x $ 1

281617

11

x $ 7

x

10

10x

8

873

34

x

69

12x

x

36

1820

Skills PracticeParts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


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Find the perimeter of the given triangle.

1. "JKL, if "JKL # "RST, RS # 14, 2. "DEF, if "DEF # "ABC, AB # 27,ST # 12, TR # 10, and LJ # 14 BC # 16, CA # 25, and FD # 15

3. "PQR, if "PQR # "LMN, LM # 16, 4. "KLM, if "KLM # "FGH, FG # 30,MN # 14, NL # 27, and RP # 18 GH # 38, HF # 38, and KL # 24


5. Find FG if "RST # "EFG, S!H! is an 6. Find MN if "ABC # "MNP, A!D! is an altitude of "RST, F!J! is an altitude of altitude of "ABC, M!Q! is an altitude of"EFG, ST # 6, SH # 5, and FJ # 7. "MNP, AB # 24, AD # 14, and MQ # 10.5.

Find x.

7. "HKL # "XYZ 8.

LH

K

18

10

Y

15

x

ZX7

20

24

x

C

A

BD P NQ

M

J GE

F

H TR

S

GL

K

M

F

H

L MP Q

RN

A

B

C

D

E

F

R S

T

J K

L


Find the perimeter of the given triangle.

1. "DEF, if "ABC # "DEF, AB # 36, 2. "STU, if "STU # "KLM, KL # 12,BC # 20, CA # 40, and DE # 35 LM # 31, MK # 32, and US # 28


3. Find PR if "JKL # "NPR, K!M! is an 4. Find ZY if "STU # "XYZ, U!A! is an altitude of "JKL, P!T! is an altitude of altitude of "STU, Z!B! is an altitude of "NPR, KL # 28, KM # 18, and "XYZ, UT # 8.5, UA # 6, and PT # 15.75. ZB # 11.4.

Find x.

5. 6.

PHOTOGRAPHY For Exercises 7 and 8, use the following information.Francine has a camera in which the distance from the lens to the film is 24 millimeters.

7. If Francine takes a full-length photograph of her friend from a distance of 3 meters andthe height of her friend is 140 centimeters, what will be the height of the image on thefilm? (Hint: Convert to the same unit of measure.)

8. Suppose the height of the image on the film of her friend is 15 millimeters. If Francinetook a full-length shot, what was the distance between the camera and her friend?

x28

20 30402x $ 1 25x $ 4

YX B

Z

TS A

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MJ L

K

TN R

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CFDA

Practice Parts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

Reading to Learn MathematicsParts of Similar Triangles

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5


Less

on

6-5

Pre-Activity How is geometry related to photography?


• How is similarity involved in the process of making a photographic printfrom a negative?

• Why do photographers place their cameras on tripods?

Reading the Lesson

1. In the figure, "RST # "UVW. Complete each proportion involving the lengths of segments inthis figure by replacing the question mark. Thenidentify the definition or theorem from the listbelow that the completed proportion illustrates.

i. Definition of congruent polygons ii. Definition of similar polygons

iii. Proportional Perimeters Theorem iv. Angle Bisectors Theorem

v. Similar triangles have corresponding altitudes proportional to corresponding sides.

vi. Similar triangles have corresponding medians proportional to corresponding sides.

vii. Similar triangles have corresponding angle bisectors proportional to corresponding sides.

a. !RS % S?T % TR! # !U

RSV! b. !U

RWT! # !

S?X!

c. !RU

MN! # !V

?W! d. !U

RSV! # !

S?T!

e. !RPS

P! # !S

?T! f. !

U?N! # !

UR

WT!

g. !WTP

Q! # !R?T! h. !

UVW

W! # !Q

?V!


2. A good way to remember a large amount of information is to remember key words. Whatkey words will help you remember the features of similar triangles that are proportionalto the lengths of the corresponding sides?

W

Q N

YU

V

T

P M

XR

S


Proportions for Similar TrianglesRecall that if a line crosses two sides of a triangle and is parallel to the thirdside, then the line separates the two sides that it crosses into segments ofproportional lengths.

You can write many proportions by identifying similar triangles in thefollowing diagram. In the diagram, AM!"# & BN!"#, AE!"# & FL!"# & MR!"#, and DQ!"# & ER!"#.

Answer each question. Use the diagram above.

1. Name a triangle similar to "GNP. 2. Name a triangle similar to "CJH.

3. Name two triangles similar to "JKS. 4. Name a triangle similar to "ACP.

Complete each proportion.

5. !AA

GP! # !

A?F! 6. !C

CHP! # !

C?R! 7. !J

JRS! # !J

?L!

8. !PP

HC! # !

P?G! 9. !

ELR

R! # !J

?R! 10. !

MM

NP! # !A

?P!

Solve.

11. If CJ # 16, JR # 48, and LR # 30, find EL.

12. If DK # 5, KS # 7, and CJ # 8, find JS.

13. If MN # 12, NP # 32, and AP # 48, find AG. Round to the nearest tenth.

14. If CH # 18, HP # 82, and CR # 130, find CJ.

15. Write three more problems that can be solved using the diagram above.

A B C D E

F G H J K L

M N P RQ

S

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-56-5

Study Guide and InterventionFractals and Self-Similarity

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6


Less

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Characteristics of Fractals The act of repeating a process over and over, such asfinding a third of a segment, then a third of the new segment, and so on, is called iteration.When the process of iteration is applied to some geometric figures, the results are calledfractals. For objects such as fractals, when a portion of the object has the same shape orcharacteristics as the entire object, the object can be called self-similar.

In the diagram at the right, notice that the details at each stage are similar to the details at Stage 1.

1. Follow the iteration process below to produce a fractal.

Stage 1• Draw a square.• Draw an isosceles right triangle on the top side of the square.

Use the side of the square as the hypotenuse of the triangle.• Draw a square on each leg of the right triangle.

Stage 2Repeat the steps in Stage 1, drawing an isosceles triangle and two small squares for each of the small squares from Stage 1.

Stage 3Repeat the steps in Stage 1 for each of the smallest squares in Stage 2.

2. Is the figure produced in Stage 3 self-similar?

Stage 1 Stage 2 Stage 3

ExercisesExercises

ExampleExample


Nongeometric Iteration An iterative process can be applied to an algebraicexpression or equation. The result is called a recursive formula.

Find the value of x3, where the initial value of x is 2. Repeat theprocess three times and describe the pattern.

Initial value: 2First time: 23 # 8Second time: 83 # 512Third time: 5123 # 134,217,728

The result of each step of the iteration is used for the next step. For this example, the xvalues are greater with each iteration. There is no maximum value, so the values aredescribed as approaching infinity.

For Exercises 1–5, find the value of each expression. Then use that value as thenext x in the expression. Repeat the process three times, and describe yourobservations.

1. $x!, where x initially equals 5

2. !1x!, where x initially equals 2

3. 3x, where x initially equals 1

4. x & 5 where x initially equals 10

5. x2 & 4, where x initially equals 16

6. Harpesh paid $1000 for a savings certificate. It earns interest at an annual rate of 2.8%,and interest is added to the certificate each year. What will the certificate be worth afterfour years?


Fractals and Self-Similarity

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

ExercisesExercises

ExampleExample

Skills PracticeFractals and Self-Similarity

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6


Less

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Stages 1 and 2 of a fractal known as the Cantor set are shown. To get Stage 2, thesegment in Stage 1 is trisected, and the interior of the middle segment is removed.(The interior of a segment is the segment with its endpoints removed.) Theprocess is repeated for subsequent stages.

1. Draw stages 3 and 4 of the Cantor set.

2. How many segments are there in Stage 3? Stage 4?

3. What happens to the length of the line segments in each stage?

4. The Cantor set is the set of points after infinitely many iterations. Is the Cantor set self-similar?

Find the value of each expression. Then, use that value as the next x in theexpression. Repeat the process three times, and describe your observations.

5. 2x & 3, where x initially equals 5

6. !2x

!, where x initially equals 1

Find the first three iterates of each expression.

7. x2 & 1, where x initially equals 2 8. !1x!, where x initially equals 8

Stage 3

Stage 4

Stage 1

Stage 2


An artist is designing a book cover and wants to show a copy of the book cover inthe lower left corner of the cover. After Stages 1 and 2 of the design, he realizesthat the design is developing into a fractal!

1. Draw Stages 3 and 4 of the book cover fractal.

2. After infinitely many iterations, will the result be a self similar fractal?

3. On what part of the book cover should you focus your attention to be sure you can find acopy of the entire figure?

Find the value of each expression. Then, use that value as the next x in theexpression. Repeat the process three times and describe your observations.

4. 2(x % 3), where x initially equals 0 5. !2x

! & 3, where x initially equals 10

Find the first three iterates of each expression.

6. !2x

! % 1, where x initially equals 1 7. 2x2, where x initially equals 2

8. HOUSING The Andrews purchased a house for $96,000. The real estate agent who soldthe house said that comparable houses in the area appreciate at a rate of 4.5% per year.If this pattern continues, what will be the value of the house in three years? Round tothe nearest whole number.

Stage 3

MATHMATH

MATH

Stage 4

MATHMATH

MATHMATH

Stage 1

MATHStage 2

MATHMATH

Practice Fractals and Self-Similarity

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

Reading to Learn MathematicsFractals and Self-Similarity

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6


Less

on

6-6

Pre-Activity How is mathematics found in nature?


Name two objects from nature other than broccoli in which a small pieceresembles the whole.

Reading the Lesson1. Match each definition from the first column with a term from the second column. (Some

words of phrases in the second column may be used more than once or not at all.)

Phrase Terma. a geometric figure that is created using iteration i. similar b. a pattern in which smaller and smaller details of a shape have ii. iteration

the same geometric characteristics as the original shape iii. fractalc. the result of translating an iterative process into a formula or iv. self-similar

algebraic equation v. recursiond. the process of repeating the same procedure over and over formula

again vi. congruent

2. Refer to the three patterns below.

i.

ii.

iii.

a. Give the special name for each of the fractals you obtain by continuing the patternwithout end.

b. Which of these fractals are self-similar?c. Which of these fractals are strictly self-similar?

Helping You Remember3. A good way to remember a new mathematical term is to relate it to everyday English

words. The word fractal is related to fraction and fragment. Use your dictionary to findat least one definition for each of these words that you think is related to the meaning ofthe word fractal and explain the connection.


The Möbius StripA Möbius strip is a special surface with only one side. It was discovered byAugust Ferdinand Möbius, a German astronomer and mathematician.

1. To make a Möbius strip, cut a strip of paper about 16 inches long and 1inch wide. Mark the ends with the letters A, B, C, and D as shown below.

Twist the paper once, connecting A to D and B to C. Tape the endstogether on both sides.

2. Use a crayon or pencil to shade one side of the paper. Shade around thestrip until you get back to where you started. What happens?

3. What do you think will happen if you cut the Möbius strip down themiddle? Try it.

4. Make another Möbius strip. Starting a third of the way in from one edge,cut around the strip, staying always the same distance in from the edge.What happens?

5. Start with another long strip of paper. Twist the paper twice and connectthe ends. What happens when you cut down the center of this strip?

6. Start with another long strip of paper. Twist the paper three times andconnect the ends. What happens when you cut down the center of this strip?

A

B

C

D

Enrichment

NAME ______________________________________________ DATE ____________ PERIOD _____

6-66-6

© Glencoe/McGraw-Hill A2 Glencoe Geometry

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162

.W

rite

a r

atio

for

th

e n

um

ber

of g

ames

won

to

the

tota

l n

um

ber

of g

ames

pla

yed

.To

fin

d th

e ra

tio,

divi

de t

he n

umbe

r of

gam

es w

on b

y th

e to

tal n

umbe

r of

gam

es p

laye

d.T

he

resu

lt is

! 16 67 2!,w

hich

is a

bout

0.4

1.T

he C

hica

go C

ubs

won

abo

ut 4

1% o

f th

eir

gam

es in

200

2.

A d

oll

hou

se t

hat

is

15 i

nch

es t

all

is a

sca

le m

odel

of

a re

al h

ouse

wit

h a

hei

ght

of 2

0 fe

et.W

hat

is

the

rati

o of

th

e h

eigh

t of

th

e d

oll

hou

se t

o th

eh

eigh

t of

th

e re

al h

ouse

?To

sta

rt,c

onve

rt t

he h

eigh

t of

the

rea

l hou

se t

o in

ches

.20

fee

t "

12 in

ches

per

foo

t #

240

inch

esTo

find

the

rat

io o

r sc

ale

fact

or o

f th

e he

ight

s,di

vide

the

hei

ght

of t

he d

oll h

ouse

by

the

heig

ht o

f th

e re

al h

ouse

.The

rat

io is

15

inch

es:2

40 in

ches

or

1:16

.The

hei

ght

of t

he d

oll

hous

e is

! 11 6!th

e he

ight

of

the

real

hou

se.

1.In

the

200

2 M

ajor

Lea

gue

base

ball

seas

on,S

amm

y So

sa h

it 4

9 ho

me

runs

and

was

at

bat

556

tim

es.F

ind

the

rati

o of

hom

e ru

ns t

o th

e nu

mbe

r of

tim

es h

e w

as a

t ba

t.

! 54 59 6!

2.T

here

are

182

gir

ls in

the

sop

hom

ore

clas

s of

305

stu

dent

s.F

ind

the

rati

o of

gir

ls t

o to

tal s

tude

nts.

!1 38 02 5!

3.T

he le

ngth

of

a re

ctan

gle

is 8

inch

es a

nd it

s w

idth

is 5

inch

es.F

ind

the

rati

o of

leng

th

to w

idth

.

!8 5!

4.T

he s

ides

of

a tr

iang

le a

re 3

inch

es,4

inch

es,a

nd 5

inch

es.F

ind

the

scal

e fa

ctor

bet

wee

nth

e lo

nges

t an

d th

e sh

orte

st s

ides

.

!5 3!

5.T

he le

ngth

of

a m

odel

tra

in is

18

inch

es.I

t is

a s

cale

mod

el o

f a

trai

n th

at is

48

feet

long

.F

ind

the

scal

e fa

ctor

.

! 31 2!

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exercis

esExercis

es

©G

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eom

etry

Use

Pro

per

ties

of

Pro

po

rtio

ns

A s

tate

men

t th

at t

wo

rati

os a

re e

qual

is c

alle

d a

pro

por

tion

.In

the

prop

orti

on !a b!

#! dc ! ,

whe

re b

and

dar

e no

t ze

ro,t

he v

alue

s a

and

dar

e th

e ex

trem

esan

d th

e va

lues

ban

d c

are

the

mea

ns.

In a

pro

port

ion,

the

prod

uct

of t

hem

eans

is e

qual

to

the

prod

uct

of t

he e

xtre

mes

,so

ad#

bc.

!a b!#

! dc !

a$

d#

b$

c↑

↑ex

trem

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eans

Sol

ve ! 19 6!

"!2 x7 !

.

! 19 6!#

!2 x7 !

9 $

x#

16 $

27C

ross

pro

duct

s

9x#

432

Mul

tiply

.

x#

48D

ivid

e ea

ch s

ide

by 9

.

A r

oom

is

49 c

enti

met

ers

by 2

8 ce

nti

met

ers

on a

sca

le d

raw

ing

of a

hou

se.F

or t

he

actu

al r

oom

,th

e la

rger

dim

ensi

on i

s 14

fee

t.F

ind

th

e sh

orte

rd

imen

sion

of

the

actu

al r

oom

.If

xis

the

roo

m’s

sho

rter

dim

ensi

on,t

hen

!2 48 9!#

! 1x 4!

49x

#39

2C

ross

pro

duct

s

x#

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ivid

e ea

ch s

ide

by 4

9.

The

sho

rter

sid

e of

the

roo

m is

8 f

eet.

Sol

ve e

ach

pro

por

tion

.

1.!1 2!

#!2 x8 !

562.

!3 8!#

! 2y 4!9

3.!x x% %

2 22!

#!3 10 0!

8

4.! 18

3 .2!#

!9 y!54

.65.

!2x8%

3!

#!5 4!

3.5

6.!x x

% &1 1

!#

!3 4!#

7

Use

a p

rop

orti

on t

o so

lve

each

pro

blem

.

7.If

3 c

asse

ttes

cos

t $4

4.85

,fin

d th

e co

st o

f on

e ca

sset

te.

$14.

95

8.T

he r

atio

of

the

side

s of

a t

rian

gle

are

8:15

:17.

If t

he p

erim

eter

of

the

tria

ngle

is

480

inch

es,f

ind

the

leng

th o

f ea

ch s

ide

of t

he t

rian

gle.

96 in

.,18

0 in

.,20

4 in

.

9.T

he s

cale

on

a m

ap in

dica

tes

that

one

inch

equ

als

4 m

iles.

If t

wo

tow

ns a

re 3

.5 in

ches

apar

t on

the

map

,wha

t is

the

act

ual d

ista

nce

betw

een

the

tow

ns?

14 m

i

shor

ter

dim

ensi

on!

!!

long

er d

imen

sion

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Pro

port

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-1

6-1

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exercis

esExercis

es

Answers (Lesson 6-1)


An

swer

s

Skil

ls P

ract

ice

Pro

port

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-1

6-1

©G

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7G

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etry

Lesson 6-1

1.FO

OTB

ALL

A t

ight

end

sco

red

6 to

uchd

owns

in 1

4 ga

mes

.Fin

d th

e ra

tio

of t

ouch

dow

nspe

r ga

me.

0.43

:1

2.E

DU

CA

TIO

NIn

a s

ched

ule

of 6

cla

sses

,Mar

ta h

as 2

ele

ctiv

e cl

asse

s.W

hat

is t

he r

atio

of e

lect

ive

to n

on-e

lect

ive

clas

ses

in M

arta

’s s

ched

ule?

1:2

3.B

IOLO

GY

Out

of

274

liste

d sp

ecie

s of

bir

ds in

the

Uni

ted

Stat

es,7

8 sp

ecie

s m

ade

the

enda

nger

ed li

st.F

ind

the

rati

o of

end

ange

red

spec

ies

of b

irds

to

liste

d sp

ecie

s in

the

Uni

ted

Stat

es.

! 13 39 7!

4.A

RTA

n ar

tist

in P

ortl

and,

Ore

gon,

mak

es b

ronz

e sc

ulpt

ures

of

dogs

.The

rat

io o

f th

ehe

ight

of

a sc

ulpt

ure

to t

he a

ctua

l hei

ght

of t

he d

og is

2:3

.If

the

heig

ht o

f th

e sc

ulpt

ure

is 1

4 in

ches

,fin

d th

e he

ight

of

the

dog.

21 in

.

5.SC

HO

OL

The

rat

io o

f m

ale

stud

ents

to

fem

ale

stud

ents

in t

he d

ram

a cl

ub a

t C

ampb

ell

Hig

h Sc

hool

is 3

:4.I

f th

e nu

mbe

r of

mal

e st

uden

ts in

the

clu

b is

18,

wha

t is

the

num

ber

of f

emal

e st

uden

ts?

24

Sol

ve e

ach

pro

por

tion

.

6.!2 5!

#! 4x 0!

167.

! 17 0!#

!2 x1 !30

8.!2 50 !

#!4 6x !

6

9.!5 4x !

#!3 85 !

3.5

10.!

x% 3

1!

#!7 2!

9.5

11.!

1 35 !#

!x& 5

3!

28

Fin

d t

he

mea

sure

s of

th

e si

des

of

each

tri

angl

e.

12.T

he r

atio

of

the

mea

sure

s of

the

sid

es o

f a

tria

ngle

is 3

:5:7

,and

its

peri

met

er is

45

0 ce

ntim

eter

s.90

cm

,150

cm

,210

cm

13.T

he r

atio

of t

he m

easu

res

of t

he s

ides

of a

tri

angl

e is

5:6

:9,a

nd it

s pe

rim

eter

is 2

20 m

eter

s.55

m,6

6 m

,99

m

14.T

he r

atio

of

the

mea

sure

s of

the

sid

es o

f a

tria

ngle

is 4

:6:8

,and

its

peri

met

er is

126

fee

t.28

ft,

42 f

t,56

ft

15.T

he r

atio

of t

he m

easu

res

of t

he s

ides

of a

tri

angl

e is

5:7

:8,a

nd it

s pe

rim

eter

is 4

0 in

ches

.10

in.,

14 in

.,16

in.

©G

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ill29

8G

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eom

etry

1.N

UTR

ITIO

NO

ne o

unce

of c

hedd

ar c

hees

e co

ntai

ns 9

gra

ms

of fa

t.Si

x of

the

gra

ms

of fa

tar

e sa

tura

ted

fats

.Fin

d th

e ra

tio

of s

atur

ated

fats

to

tota

l fat

in a

n ou

nce

of c

hees

e.2:

3

2.FA

RM

ING

The

rat

io o

f go

ats

to s

heep

at

a un

iver

sity

res

earc

h fa

rm is

4:7

.The

num

ber

of s

heep

at

the

farm

is 2

8.W

hat

is t

he n

umbe

r of

goa

ts?

16

3.A

RTE

dwar

d H

oppe

r’s o

il on

can

vas

pain

ting

Nig

htha

wks

has

a le

ngth

of

60 in

ches

and

a w

idth

of

30 in

ches

.A p

rint

of

the

orig

inal

has

a le

ngth

of

2.5

inch

es.W

hat

is t

he w

idth

of t

he p

rint

?1.

25 in

.

Sol

ve e

ach

pro

por

tion

.

4.!5 8!

#! 1x 2!

7.5

5.! 1.

x 12!#

!1 5!0.

224

6.! 26 7x !

#!4 3!

6

7.!x

% 32

!#

!8 9!!2 3!

8.!3x

4&5

!#

!& 75 !!5 7!

9.!x

& 42

!#

!x% 2

4!

#10

Fin

d t

he

mea

sure

s of

th

e si

des

of

each

tri

angl

e.

10.T

he r

atio

of

the

mea

sure

s of

the

sid

es o

f a

tria

ngle

is 3

:4:6

,and

its

peri

met

er is

104

fee

t.24

ft,

32 f

t,48

ft

11.T

he r

atio

of t

he m

easu

res

of t

he s

ides

of a

tri

angl

e is

7:9

:12,

and

its

peri

met

er is

84

inch

es.

21 in

.,27

in.,

36 in

.

12.T

he r

atio

of

the

mea

sure

s of

the

sid

es o

f a

tria

ngle

is 6

:7:9

,and

its

peri

met

er is

77

cen

tim

eter

s.21

cm

,24.

5 cm

,31.

5 cm

Fin

d t

he

mea

sure

s of

th

e an

gles

in

eac

h t

rian

gle.

13.T

he r

atio

of

the

mea

sure

s of

the

ang

les

is 4

:5:6

.48

,60,

72

14.T

he r

atio

of

the

mea

sure

s of

the

ang

les

is 5

:7:8

.45

,63,

72

15.B

RID

GES

The

spa

n of

the

Ben

jam

in F

rank

lin s

uspe

nsio

n br

idge

in P

hila

delp

hia,

Penn

sylv

ania

,is

1750

fee

t.A

mod

el o

f th

e br

idge

has

a s

pan

of 4

2 in

ches

.Wha

t is

the

rati

o of

the

spa

n of

the

mod

el t

o th

e sp

an o

f th

e ac

tual

Ben

jam

in F

rank

lin B

ridg

e?! 51 00!

Pra

ctic

e (A

vera

ge)

Pro

port

ions

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-1

6-1



Rea

din

g t

o L

earn

Math

emati

csP

ropo

rtio

ns

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-1

6-1

©G

lenc

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ill29

9G

lenc

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etry

Lesson 6-1

Pre-

Act

ivit

yH

ow d

o ar

tist

s u

se r

atio

s?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

1 at

the

top

of

page

282

in y

our

text

book

.

Est

imat

e th

e ra

tio

of le

ngth

to

wid

th f

or t

he b

ackg

roun

d re

ctan

gles

inT

iffa

ny’s

Cle

mat

is S

kylig

ht.

Sam

ple

answ

er:a

bout

5 t

o 2

Rea

din

g t

he

Less

on

1.M

atch

eac

h de

scri

ptio

n in

the

fir

st c

olum

n w

ith

a w

ord

or p

hras

e fr

om t

he s

econ

dco

lum

n.

a.T

he r

atio

of

two

corr

espo

ndin

g qu

anti

ties

ivi.

prop

orti

on

b.r

and

uin

the

equ

atio

n !r s!

#! ut !

vii

.cro

ss p

rodu

cts

c.a

com

pari

son

of t

wo

quan

titi

esvi

iii.

mea

ns

d.ru

and

stin

the

equ

atio

n !r s!

#! ut !

iiiv

.sc

ale

fact

or

e.an

equ

atio

n st

atin

g th

at t

wo

rati

os a

re e

qual

iv.

extr

emes

f.s

and

tin

the

equ

atio

n !r s!

#! ut !

iiivi

.rat

io

2.If

m,n

,p,a

nd q

are

nonz

ero

num

bers

suc

h th

at !m n!

#!p q! ,

whi

ch o

f th

e fo

llow

ing

stat

emen

ts c

ould

be

fals

e?B

,C,E

A.n

p#

mq

B.!

np !#

! mq !

C.m

p#

nqD

.qm

#pn

E.!

m n!#

! pq !F.

! pq !#

! mn !

G.m

:p#

n:q

H.m

:n#

p:q

3.W

rite

tw

o pr

opor

tion

s th

at m

atch

eac

h de

scri

ptio

n.

a.M

eans

are

5 a

nd 8

;ext

rem

es a

re 4

and

10.

any

two

of !4 5!

"! 18 0!

,!4 8!

"! 15 0!

,!1 50 !

"!8 4! ,

!1 80 !"

!5 4!

b.M

eans

are

5 a

nd 4

;ext

rem

es a

re p

osit

ive

inte

gers

tha

t ar

e di

ffer

ent

from

mea

ns.

any

two

of !2 5!

"! 14 0!

,!2 4!

"! 15 0!

,!1 4!

"! 25 0!

,!1 5!

"! 24 0!

,!1 50 !

"!4 2! ,

!1 40 !"

!5 2! ,!2 40 !

"!5 1! ,

or !2 50 !

"!4 1!

Hel

pin

g Y

ou

Rem

emb

er

4.So

met

imes

it is

eas

ier

to r

emem

ber

a m

athe

mat

ical

idea

if y

ou p

ut it

into

wor

ds w

itho

utus

ing

any

mat

hem

atic

al s

ymbo

ls.H

ow c

an y

ou u

se t

his

appr

oach

to

rem

embe

r th

eco

ncep

t of

equ

alit

y of

cro

ss p

rodu

cts?

Sam

ple

answ

er:T

he p

rodu

ct o

f th

e m

eans

equa

ls t

he p

rodu

ct o

f th

e ex

trem

es.

©G

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ill30

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etry

Gol

den

Rec

tang

les

Use

a s

trai

ghte

dge

,com

pas

s,an

d t

he

inst

ruct

ion

s be

low

to c

onst

ruct

a g

old

en r

ecta

ngl

e.

1.C

onst

ruct

squ

are

AB

CD

wit

h si

des

of 2

cm

.

2.C

onst

ruct

the

mid

poin

t of

A !B!

.Cal

l the

mid

poin

t M

.

3.D

raw

AB

! "# .

Set

your

com

pass

at

an o

peni

ng e

qual

to

MC

.U

se M

as t

he c

ente

r to

dra

w a

n ar

c th

at in

ters

ects

AB

! "# .

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idth

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y fi

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e va

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sai

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plea

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igu

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gol

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6-2

6-2



An

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Pre-

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Rea

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n 6-

2 at

the

top

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page

289

in y

our

text

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Des

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gure

s th

at h

ave

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ampl

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r fe

et p

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cent

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me

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t ha

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s to

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as

your

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the

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ter

to t

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ter

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ler.

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Less

on

1.C

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Tw

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re s

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r if

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ir c

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les

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and

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r co

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s ar

e .

d.T

wo

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nt if

the

ir c

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les

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r co

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e .

e.T

he r

atio

of

the

leng

ths

of c

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es o

f tw

o si

mila

r fi

gure

s is

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led

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e in

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e by

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ctor

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r fi

gure

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alle

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If t

wo

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h a

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e fa

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n th

e po

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re

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2.D

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min

e w

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emen

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ays,

som

etim

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ver

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.a.

Tw

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imes

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Tw

o re

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etim

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osce

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t tr

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isos

cele

s ri

ght

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e si

mila

r.al

way

sg.

A s

quar

e is

sim

ilar

to a

n eq

uila

tera

l tri

angl

e.ne

ver

h.

Tw

o ac

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ngle

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e si

mila

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#"

UV

W,t

hen

"U

VW

#"

RS

T.

sym

met

ric

b.If

"R

ST

#"

UV

Wan

d "

UV

W#

"O

PQ

,the

n "

RS

T#

"O

PQ

.tr

ansi

tive

c."

RS

T#

"R

ST

refle

xive

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r so

met

hing

is t

o ex

plai

n it

to

som

eone

els

e.Su

ppos

e on

e of

you

rcl

assm

ates

is h

avin

g tr

oubl

e un

ders

tand

ing

the

diff

eren

ce b

etw

een

SAS

for

cong

ruen

ttr

iang

les

and

SAS

for

sim

ilar

tria

ngle

s.H

ow c

an y

ou e

xpla

in t

he d

iffe

renc

e to

him

?S

ampl

e an

swer

:In

SA

S fo

r co

ngru

ence

,cor

resp

ondi

ng s

ides

are

cong

ruen

t,w

hile

in S

AS

for

sim

ilari

ty,c

orre

spon

ding

sid

es a

repr

opor

tiona

l.(In

bot

h,th

e in

clud

ed a

ngle

s ar

e co

ngru

ent.)

©G

lenc

oe/M

cGra

w-H

ill31

2G

lenc

oe G

eom

etry

Rat

io P

uzzl

es w

ith T

rian

gles

If y

ou k

now

the

per

imet

er o

f a

tria

ngle

and

the

rat

ios

of t

he s

ides

,you

can

find

the

leng

ths

of t

he s

ides

.

Th

e p

erim

eter

of

a tr

ian

gle

is 8

4 u

nit

s.T

he

sid

es h

ave

len

gth

s r,

s,an

d t

.Th

e ra

tio

of s

to r

is 5

:3,a

nd

th

e ra

tio

of t

to r

is2

:1.F

ind

th

e le

ngt

h o

f ea

ch s

ide.

Sinc

e bo

th r

atio

s co

ntai

n r,

rew

rite

one

or

both

rat

ios

to m

ake

rth

e sa

me.

You

can

wri

te t

he r

atio

of t

to r

as 6

:3.N

ow y

ou c

an w

rite

a t

hree

-par

t ra

tio.

r:s:

t#

3:5

:6

The

re is

a n

umbe

r x

such

tha

t r

#3

x,s

#5

x,an

d t

#6

x.Si

nce

you

know

the

peri

met

er,8

4,yo

u ca

n us

e al

gebr

a to

fin

d th

e le

ngth

s of

the

sid

es.

r%

s%

t#

843

x%

5x%

6x#

8414

x#

84x

#6

3x

#18

,5x

#30

,6x

#36

So r

#18

,s#

30,a

nd t

#36

.

Fin

d t

he

len

gth

s of

th

e si

des

of

each

tri

angl

e.

1.T

he p

erim

eter

of

a tr

iang

le is

75

unit

s.T

he s

ides

hav

e le

ngth

s a,

b,an

d c.

The

ra

tio

of b

to a

is 3

:5,a

nd t

he r

atio

of c

to a

is 7

:5.F

ind

the

leng

th o

f ea

ch s

ide.

a"

25,b

"15

,c"

35

2.T

he p

erim

eter

of

a tr

iang

le is

88

unit

s.T

he s

ides

hav

e le

ngth

s d,

e,an

d f.

The

ra

tio

of e

to d

is 3

:1,a

nd t

he r

atio

of f

to e

is 1

0:9.

Fin

d th

e le

ngth

of

each

sid

e.

d"

12,e

"36

,f"

40

3.T

he p

erim

eter

of

a tr

iang

le is

91

unit

s.T

he s

ides

hav

e le

ngth

s p,

q,an

d r.

The

ra

tio

of p

to r

is 3

:1,a

nd t

he r

atio

of q

to r

is 5

:2.F

ind

the

leng

th o

f ea

ch s

ide.

p"

42,q

"35

,r"

14

4.T

he p

erim

eter

of

a tr

iang

le is

68

unit

s.T

he s

ides

hav

e le

ngth

s g,

h,an

d j.

The

ra

tio

of j

to g

is 2

:1,a

nd t

he r

atio

of h

to g

is 5

:4.F

ind

the

leng

th o

f ea

ch s

ide.

g"

16,h

"20

,j"

32

5.W

rite

a p

robl

em s

imila

r to

tho

se a

bove

invo

lvin

g ra

tios

in t

rian

gles

.

See

stu

dent

s’w

ork.

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-3

6-3

Exam

ple

Exam

ple



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Para

llel L

ines

and

Pro

port

iona

l Par

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill31

3G

lenc

oe G

eom

etry

Lesson 6-4

Pro

po

rtio

nal

Par

ts o

f Tr

ian

gle

sIn

any

tri

angl

e,a

line

para

llel t

o on

e si

de o

f a

tria

ngle

sep

arat

es t

he o

ther

tw

o si

des

prop

orti

onal

ly.T

he c

onve

rse

is a

lso

true

.

If X

and

Yar

e th

e m

idpo

ints

of R !

T!an

d S!T!

,the

n X!

Y!is

am

idse

gmen

tof

the

tri

angl

e.T

he T

rian

gle

Mid

segm

ent

The

orem

stat

es t

hat

a m

idse

gmen

t is

par

alle

l to

the

thir

d si

de a

nd is

hal

f it

s le

ngth

.

If X

Y! "

#|| R

S!"

# ,th

en !R X

TX !#

! YSY T!

.

If !R X

TX !#

! YSY T!

,the

n X

Y!"

#|| R

S!"

# .

If X!

Y!is

a m

idse

gmen

t,th

en X

Y!"

#|| R

S!"

#an

d X

Y#

!1 2! RS

.

YS

T

R

X

In !

AB

C,E"

F"|| C"

B".F

ind

x.

Sinc

e E!

F!|| C!

B!,!

FABF !

#! EA

E C!.

!x x% %2 22

!#

!1 68 !

6x%

132

#18

x%

3696

#12

x8

#x

F

C

BA

18

x %

22

x %

2

6E

A t

rian

gle

has

ver

tice

sD

(3,6

),E

(#3,

#2)

,an

d F

(7,#

2).

Mid

segm

ent

G "H"

is p

aral

lel

to E"

F".F

ind

th

e le

ngt

h o

f G "

H".

G !H!

is a

mid

segm

ent,

so it

s le

ngth

is o

ne-

half

tha

t of

E !F!.

Poin

ts E

and

Fha

ve t

hesa

me

y-co

ordi

nate

,so

EF

#7

&(&

3) #

10.

The

leng

th o

f m

idse

gmen

t G !

H!is

5.

Exam

ple1

Exam

ple1

Exam

ple2

Exam

ple2

Exercis

esExercis

es

Fin

d x

.

1.7

2.10

3.17

.5

4.8

5.12

6.5

7.In

Exa

mpl

e 2,

find

the

slo

pe o

f E!F!

and

show

tha

tE!

F!|| G!

H!.

The

endp

oint

s of

G"H"

are

at (

0,2)

and

(5,

2).T

he s

lope

of

G"H"

is 0

and

the

sl

ope

of E"

F"is

0.G"

H"an

d E"F"

are

para

llel.

3010

x %

10

x11

x %

12

x33

3010

24x

35x9

20

x

185

5x

7

©G

lenc

oe/M

cGra

w-H

ill31

4G

lenc

oe G

eom

etry

Div

ide

Seg

men

ts P

rop

ort

ion

ally

Whe

n th

ree

or m

ore

para

llel l

ines

cut

tw

o tr

ansv

ersa

ls,

they

sep

arat

e th

e tr

ansv

ersa

ls in

to p

ropo

rtio

nal

part

s.If

the

rat

io o

f th

e pa

rts

is 1

,the

n th

e pa

ralle

llin

es s

epar

ate

the

tran

sver

sals

into

con

grue

nt p

arts

.

If !

1|| !

2|| !

3,If

!4

|| !5

|| !6

and

then

!a b!#

! dc ! .!u v!

#1,

then

!w x!#

1.

Ref

er t

o li

nes

!1,

! 2,a

nd

!3

abov

e.If

a"

3,b

"8,

and

c"

5,fi

nd

d.

! 1|| !

2|| !

3so

!3 8!#

! d5 ! .T

hen

3d#

40 a

nd d

#13

!1 3! .

Fin

d x

and

y.

1.2.

x"

8x

"9

3.4.

x"

5;y

"2

y"

3!1 3!

5.6.

x"

12y

"3

1632

y %

3y

34

x %

4x

5

8y

% 2

y2x

% 4

3 x #

12y

% 2

3 y

2x #

6

x %

3

12

x %

12

123x5x

a

bd

uv x

w

ct

n m

s! 1

! 4! 5

! 6

! 2

! 3

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Para

llel L

ines

and

Pro

port

iona

l Par

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-4

6-4

Exam

ple

Exam

ple

Exercis

esExercis

es



Skil

ls P

ract

ice

Para

llel L

ines

and

Pro

port

iona

l Par

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill31

5G

lenc

oe G

eom

etry

Lesson 6-4

1.If

JK

#7,

KH

#21

,and

JL

#6,

2.F

ind

xan

d T

Vif

RU

#8,

US

#14

,fi

nd L

I.T

V#

x&

1 an

d V

S#

17.5

.

1811

;10

Det

erm

ine

wh

eth

er B!

C!|| D!

E!.

3.A

D#

15,D

B#

12,A

E#

10,a

nd E

C#

8ye

s

4.B

D#

9,B

A#

27,a

nd C

Eis

one

thi

rd o

f EA

no

5.A

E#

30,A

C#

45,a

nd A

Dis

tw

ice

DB

yes

CO

OR

DIN

ATE

GEO

MET

RYF

or E

xerc

ises

6–8

,use

th

e fo

llow

ing

info

rmat

ion

.T

rian

gle

AB

Cha

s ve

rtic

es A

(&5,

2),B

(1,8

),an

d C

(4,2

).Po

int

Dis

the

mid

poin

t of

A !B!

and

Eis

the

mid

poin

t of

A!C!

.

6.Id

enti

fy t

he c

oord

inat

es o

f Dan

d E

.D

(#2,

5);E

(#0.

5,2)

7.Sh

ow t

hat

B!C!

is p

aral

lel t

o D!

E!.

Sin

ce th

e sl

ope

of B"

C"is

#2

and

the

slop

e of

D"E"

is #

2,B"

C"is

par

alle

l to

D"E".

8.Sh

ow t

hat

DE

#!1 2! B

C.

The

leng

th o

f D"

E"is

#11

.25

"or

an

d th

e le

ngth

of

B"C"

is #

45".

9.F

ind

xan

d y.

10.F

ind

xan

d y.

x"

6,y

"6.

5x

"2,

y"

4

3 – 2x %

2

x %

3

2 y #

1

3y #

52x

% 1

3 y #

8y

% 5

x %

7

#45"

!2

x

y

O

AC

B

D

E

CB

ED

A

R

UV

T

S

HL

KJ

I

©G

lenc

oe/M

cGra

w-H

ill31

6G

lenc

oe G

eom

etry

1.If

AD

#24

,DB

#27

,and

EB

#18

,2.

Fin

d x,

QT

,and

TR

if Q

T#

x%

6,fi

nd C

E.

SR

#12

,PS

#27

,and

TR

#x

&4.

1612

;18;

8

Det

erm

ine

wh

eth

er J"

K"|| N"

M".

3.JN

#18

,JL

#30

,KM

#21

,and

ML

#35

no

4.K

M#

24,K

L#

44,a

nd N

L#

!5 6! JN

yes

CO

OR

DIN

ATE

GEO

MET

RYF

or E

xerc

ises

5 a

nd

6,u

se t

he

foll

owin

g in

form

atio

n.

Tri

angl

e E

FG

has

vert

ices

E(&

4,&

1),F

(2,5

),an

d G

(2,&

1).P

oint

Kis

the

mid

poin

t of

E !G!

and

His

the

mid

poin

t of

F!G!

.

5.Sh

ow t

hat

E!F!

is p

aral

lel t

o K!

H!.

The

slop

e of

E"F"

is 1

and

the

slo

pe o

f K"

H"is

1,

so E"

F"is

par

alle

l to

K"H"

.

6.Sh

ow t

hat

KH

#!1 2! E

F.

The

leng

th o

f K"

H"is

3#

2",an

d th

e le

ngth

of

E"F"is

6#

2".

7.F

ind

xan

d y.

8.F

ind

xan

d y.

x"

4,y

"9

x"

!5 2! ,y

"!9 4!

9.M

APS

The

dis

tanc

e fr

om W

ilmin

gton

to

Ash

Gro

ve a

long

K

enda

ll is

820

fee

t an

d al

ong

Mag

nolia

,660

fee

t.If

the

di

stan

ce b

etw

een

Bee

ch a

nd A

sh G

rove

alo

ng M

agno

lia is

28

0 fe

et,w

hat

is t

he d

ista

nce

betw

een

the

two

stre

ets

alon

gK

enda

ll?ab

out

348

ft

Ash

Grov

e

Beec

h

Mag

nolia

Kend

all

Wilm

ingt

on

4 – 3y %

2

3 x #

4x

% 1

4 y #

43x

# 4

2 – 3y %

31 – 3y

% 6

5 – 4x %

3

x

y

O

EGF H

K

NL

J

KM

S

Q

TR

P

D

EC BA

Pra

ctic

e (A

vera

ge)

Para

llel L

ines

and

Pro

port

iona

l Par

ts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-4

6-4



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csPa

ralle

l Lin

es a

nd P

ropo

rtio

nal P

arts

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-4

6-4

©G

lenc

oe/M

cGra

w-H

ill31

7G

lenc

oe G

eom

etry

Lesson 6-4

Pre-

Act

ivit

yH

ow d

o ci

ty p

lan

ner

s u

se g

eom

etry

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

4 at

the

top

of

page

307

in y

our

text

book

.

Use

a g

eom

etri

c id

ea t

o ex

plai

n w

hy t

he d

ista

nce

betw

een

Chi

cago

Ave

nue

and

Ont

ario

Str

eet

is s

hort

er a

long

Mic

higa

n A

venu

e th

an a

long

Lak

eSh

ore

Dri

ve.S

ampl

e an

swer

:The

sho

rtes

t di

stan

ce b

etw

een

two

para

llel l

ines

is t

he p

erpe

ndic

ular

dis

tanc

e.

Rea

din

g t

he

Less

on

1.P

rovi

de t

he m

issi

ng w

ords

to

com

plet

e th

e st

atem

ent

of e

ach

theo

rem

.The

n st

ate

the

nam

e of

the

the

orem

.

a.If

a li

ne in

ters

ects

tw

o si

des

of a

tri

angl

e an

d se

para

tes

the

side

s in

to c

orre

spon

ding

segm

ents

of

leng

ths,

then

the

line

is

to t

he

thir

d si

de.

Con

vers

e of

the

Tri

angl

e P

ropo

rtio

nalit

y Th

eore

mb.

A m

idse

gmen

t of

a t

rian

gle

is

to o

ne s

ide

of t

he t

rian

gle

and

its

leng

th is

th

e le

ngth

of t

hat

side

.Tr

iang

le M

idse

gmen

t The

orem

c.If

a li

ne is

to

one

sid

e of

a t

rian

gle

and

inte

rsec

ts t

he o

ther

tw

o si

des

in

dist

inct

poi

nts,

then

it s

epar

ates

the

se s

ides

into

of p

ropo

rtio

nal l

engt

h.Tr

iang

le P

ropo

rtio

nalit

y Th

eore

m

2.R

efer

to

the

figu

re a

t th

e ri

ght.

a.N

ame

the

thre

e m

idse

gmen

ts o

f "R

ST

.U"

V",V"W"

,U"W"

b.If

RS

#8,

RU

#3,

and

TW

#5,

find

the

leng

th o

f ea

ch o

f th

em

idse

gmen

ts.

UV

"5,

VW

"3,

UW

"4

c.W

hat

is t

he p

erim

eter

of "

RS

T?

24d.

Wha

t is

the

per

imet

er o

f "U

VW

?12

e.W

hat

are

the

peri

met

ers

of "

RU

V,"

SV

W,a

nd "

TU

W?

12;1

2;12

f.H

ow a

re t

he p

erim

eter

s of

eac

h of

the

fou

r sm

all t

rian

gles

rel

ated

to

the

peri

met

er o

fth

e la

rge

tria

ngle

?Th

e pe

rim

eter

of

each

sm

all t

rian

gle

is o

ne-h

alf

the

peri

met

er o

f th

e la

rge

tria

ngle

.g.

Wou

ld t

he r

elat

ions

hip

that

you

fou

nd in

par

t f

appl

y to

any

tri

angl

e in

whi

ch t

hem

idpo

ints

of

the

thre

e si

des

are

conn

ecte

d?ye

s

Hel

pin

g Y

ou

Rem

emb

er

3.A

goo

d w

ay t

o re

mem

ber

a ne

w m

athe

mat

ical

ter

m is

to

rela

te it

to

othe

r m

athe

mat

ical

voca

bula

ry t

hat

you

alre

ady

know

.Wha

t is

an

easy

way

to

rem

embe

r th

e de

fini

tion

of

mid

segm

ent

usin

g ot

her

geom

etri

c te

rms?

Sam

ple

answ

er:A

mid

segm

enti

s a

segm

entt

hat

conn

ects

the

mid

poin

tsof

tw

o si

des

of a

tri

angl

e.R U

V W

T

S

segm

ents

two

para

llel

one-

half

para

llel

para

llel

prop

ortio

nal

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etry

Para

llel L

ines

and

Con

grue

nt P

arts

The

re is

a t

heor

em s

tati

ng t

hat

if t

hree

par

alle

l lin

es c

ut o

ff c

ongr

uent

seg

men

ts o

n on

etr

ansv

ersa

l,th

en t

hey

cut

off

cong

ruen

t se

gmen

ts o

n an

y tr

ansv

ersa

l.T

his

can

be s

how

n fo

ran

y nu

mbe

r of

par

alle

l lin

es.T

he f

ollo

win

g dr

afti

ng t

echn

ique

use

s th

is f

act

to d

ivid

e a

segm

ent

into

con

grue

nt p

arts

.

A !B!

is t

o be

sep

arat

ed in

to f

ive

cong

ruen

t pa

rts.

Thi

s ca

n be

do

ne v

ery

accu

rate

ly w

itho

ut u

sing

a r

uler

.All

that

is n

eede

d is

a c

ompa

ss a

nd a

pie

ce o

f no

tebo

ok p

aper

.

Ste

p 1

Hol

d th

e co

rner

of

a pi

ece

of n

oteb

ook

pape

r at

po

int

A.

Ste

p 2

Fro

m p

oint

A,d

raw

a s

egm

ent

alon

g th

e pa

per

that

is f

ive

spac

es lo

ng.M

ark

whe

re t

he li

nes

of t

he n

oteb

ook

pape

r m

eet

the

segm

ent.

Lab

el

the

fift

h po

int,

P.

Ste

p 3

Dra

w P!

B!.T

hrou

gh e

ach

of t

he o

ther

mar

ks

on A !

P!,co

nstr

uct

a lin

e pa

ralle

l to

B!P!.

The

po

ints

whe

re t

hese

line

s in

ters

ect

A !B!

will

di

vide

A !B!

into

fiv

e co

ngru

ent

segm

ents

.

Use

a c

omp

ass

and

a p

iece

of

not

eboo

k p

aper

to

div

ide

each

seg

men

t in

to t

he

give

n n

um

ber

of c

ongr

uen

t p

arts

.S

eest

uden

t’s w

ork.

1.si

x co

ngru

ent

part

s

2.se

ven

cong

ruen

t pa

rts

AB

AB

A

P

B

AB

P

AB

AB

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-4

6-4



Stu

dy

Gu

ide

and I

nte

rven

tion

Part

s of

Sim

ilar T

rian

gles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-5

6-5

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Lesson 6-5

Peri

met

ers

If t

wo

tria

ngle

s ar

e si

mila

r,th

eir

peri

met

ers

have

the

sam

e pr

opor

tion

as

the

corr

espo

ndin

g si

des.

If "

AB

C#

"R

ST

,the

n

!A RB S

% %B S

TC%%

RATC

!#

!A RB S!

#!B S

TC !#

! RAC T!

.

Use

th

e d

iagr

am a

bove

wit

h !

AB

C!

!R

ST

.If

AB

"24

an

d R

S"

15,

fin

d t

he

rati

o of

th

eir

per

imet

ers.

Sinc

e "

AB

C#

"R

ST

,the

rat

io o

f th

e pe

rim

eter

s of

"A

BC

and

"R

ST

is t

he s

ame

as t

hera

tio

of c

orre

spon

ding

sid

es.

The

refo

re

#!2 14 5!

#!8 5!

Eac

h p

air

of t

rian

gles

is

sim

ilar

.Fin

d t

he

per

imet

er o

f th

e in

dic

ated

tri

angl

e.

1."

XY

Z2.

"B

DE

3345

3."

AB

C4.

"X

YZ

8455

.4

5."

AB

C6.

"R

ST

5411

0

25 20

10

PT

SM

R

95

13

12

AD

E

C

B

56

8.7

13

10

XY

ZT

RS

18

202515

17 8

NB

AC

PM

36

9 11 CDE

A

B44

24

2022

10

TR

XZ

YS

peri

met

er o

f "A

BC

!!

!pe

rim

eter

of "

RS

T

AC

BS

RT

Exam

ple

Exam

ple

Exercis

esExercis

es

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eom

etry

Spec

ial S

egm

ents

of

Sim

ilar

Tria

ng

les

Whe

n tw

o tr

iang

les

are

sim

ilar,

corr

espo

ndin

g al

titu

des,

angl

e bi

sect

ors,

and

med

ians

are

pro

port

iona

l to

the

corr

espo

ndin

gsi

des.

Als

o,in

any

tri

angl

e an

ang

le b

isec

tor

sepa

rate

s th

e op

posi

te s

ide

into

seg

men

ts t

hat

have

the

sam

e ra

tio

as t

he o

ther

tw

o si

des

of t

he t

rian

gle.

Stu

dy

Gu

ide

and I

nte

rven

tion

(con

tinu

ed)

Part

s of

Sim

ilar T

rian

gles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-5

6-5

In t

he

figu

re,

!A

BC

!!

XY

Z,w

ith

an

gle

bise

ctor

s as

show

n.F

ind

x.

Sinc

e "

AB

C#

"X

YZ

,the

mea

sure

s of

the

angl

e bi

sect

ors

are

prop

orti

onal

to

the

mea

sure

s of

a p

air

of c

orre

spon

ding

sid

es.

!A XB Y!

#! YB WD !

!2 x4 !#

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W

Y

ZX

C

B

A

8x

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sect

s "

RS

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ind

x.

Sinc

e S !U!

is a

n an

gle

bise

ctor

,!R T

UU !#

!R TSS !

.

! 2x 0!#

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UT

S

R

Exam

ple1

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ple1

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ple2

Exam

ple2

Exercis

esExercis

es

Fin

d x

for

each

pai

r of

sim

ilar

tri

angl

es.

1.2.

108

3.4.

2.25

8.75

5.6.

12!5 6!

15

2x #

2x

% 1

2816

17

11

x %

7

x

10

10x

8

87

3

34

x

69

12x

x

36

1820



An

swer

s

Skil

ls P

ract

ice

Part

s of

Sim

ilar T

rian

gles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-5

6-5

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Lesson 6-5

Fin

d t

he

per

imet

er o

f th

e gi

ven

tri

angl

e.

1."

JKL

,if

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L#

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ST

,RS

#14

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"D

EF

,if "

DE

F#

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BC

,AB

#27

,S

T#

12,T

R#

10,a

nd L

J#

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nd F

D#

15

50.4

40.8

3."

PQ

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f "

PQ

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MN

,LM

#16

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LM

,if "

KL

M#

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GH

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#30

,M

N#

14,N

L#

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nd R

P#

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H#

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nd K

L#

24

3884

.8

Use

th

e gi

ven

in

form

atio

n t

o fi

nd

eac

h m

easu

re.

5.F

ind

FG

if "

RS

T#

"E

FG

,S!H!

is a

n 6.

Fin

d M

Nif

"A

BC

#"

MN

P,A!

D!is

an

alti

tude

of "

RS

T,F !

J!is

an

alti

tude

of

alti

tude

of "

AB

C,M!

Q!is

an

alti

tude

of

"E

FG

,ST

#6,

SH

#5,

and

FJ

#7.

"M

NP

,AB

#24

,AD

#14

,and

MQ

#10

.5.

8.4

18

Fin

d x

.

7."

HK

L#

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YZ

8.

8!1 3!

8.4

LH

K

18

10

Y

15

x

ZX

7

20

24

x

C

A

BD

PN

QM

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E

F

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Fin

d t

he

per

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er o

f th

e gi

ven

tri

angl

e.

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DE

F,i

f "A

BC

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DE

F,A

B#

36,

2."

ST

U,i

f "

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nd D

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65.6

25

Use

th

e gi

ven

in

form

atio

n t

o fi

nd

eac

h m

easu

re.

3.F

ind

PR

if "

JKL

#"

NP

R,K!

M!is

an

4.F

ind

ZY

if "

ST

U#

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YZ

,U!A!

is a

n al

titu

de o

f "JK

L,P !

T!is

an

alti

tude

of

alti

tude

of "

ST

U,Z!

B!is

an

alti

tude

of

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PR

,KL

#28

,KM

#18

,and

"

XY

Z,U

T#

8.5,

UA

#6,

and

PT

#15

.75.

ZB

#11

.4.

24.5

16.1

5

Fin

d x

.

5.6.

13.5

16.8

PHO

TOG

RA

PHY

For

Exe

rcis

es 7

an

d 8

,use

th

e fo

llow

ing

info

rmat

ion

.F

ranc

ine

has

a ca

mer

a in

whi

ch t

he d

ista

nce

from

the

lens

to

the

film

is 2

4 m

illim

eter

s.

7.If

Fra

ncin

e ta

kes

a fu

ll-le

ngth

pho

togr

aph

of h

er f

rien

d fr

om a

dis

tanc

e of

3 m

eter

s an

dth

e he

ight

of

her

frie

nd is

140

cen

tim

eter

s,w

hat

will

be

the

heig

ht o

f th

e im

age

on t

hefi

lm?

(Hin

t:C

onve

rt t

o th

e sa

me

unit

of

mea

sure

.)11

.2 m

m

8.Su

ppos

e th

e he

ight

of

the

imag

e on

the

film

of

her

frie

nd is

15

mill

imet

ers.

If F

ranc

ine

took

a f

ull-

leng

th s

hot,

wha

t w

as t

he d

ista

nce

betw

een

the

cam

era

and

her

frie

nd?

2.24

m

x28

2030

402x

% 1

25x

% 4

YX

BZ

TS

AU

MJ

L

K

TN

R

P

KS

LT

UM

B E

CF

DA

Pra

ctic

e (A

vera

ge)

Part

s of

Sim

ilar T

rian

gles

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-5

6-5



Rea

din

g t

o L

earn

Math

emati

csPa

rts

of S

imila

r Tri

angl

es

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-5

6-5

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3G

lenc

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eom

etry

Lesson 6-5

Pre-

Act

ivit

yH

ow i

s ge

omet

ry r

elat

ed t

o p

hot

ogra

ph

y?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

5 at

the

top

of

page

316

in y

our

text

book

.

•H

ow is

sim

ilari

ty in

volv

ed in

the

pro

cess

of

mak

ing

a ph

otog

raph

ic p

rint

from

a n

egat

ive?

Sam

ple

answ

er:T

he p

rint

is a

n en

larg

ed v

ersi

on o

f th

ene

gativ

e.Th

e im

age

on t

he p

rint

is s

imila

r to

the

imag

e on

the

nega

tive,

so w

hen

you

look

at

the

prin

t,yo

u w

ill s

eeex

actly

the

sam

e th

ings

in t

he s

ame

prop

ortio

ns a

s on

the

nega

tive,

just

larg

er.

•W

hy d

o ph

otog

raph

ers

plac

e th

eir

cam

eras

on

trip

ods?

Sam

ple

answ

er:T

hree

poi

nts

dete

rmin

e a

plan

e,so

the

thr

eefe

et o

f th

e tr

ipod

are

cop

lana

r an

d th

e tr

ipod

will

not

wob

ble.

Rea

din

g t

he

Less

on

1.In

the

fig

ure,

"R

ST

#"

UV

W.C

ompl

ete

each

pr

opor

tion

invo

lvin

g th

e le

ngth

s of

seg

men

ts in

this

fig

ure

by r

epla

cing

the

que

stio

n m

ark.

The

nid

enti

fy t

he d

efin

itio

n or

the

orem

fro

m t

he li

stbe

low

tha

t th

e co

mpl

eted

pro

port

ion

illus

trat

es.

i.D

efin

itio

n of

con

grue

nt p

olyg

ons

ii.

Def

init

ion

of s

imila

r po

lygo

ns

iii.

Pro

port

iona

l Per

imet

ers

The

orem

iv.

Ang

le B

isec

tors

The

orem

v.Si

mila

r tr

iang

les

have

cor

resp

ondi

ng a

ltit

udes

pro

port

iona

l to

corr

espo

ndin

g si

des.

vi.

Sim

ilar

tria

ngle

s ha

ve c

orre

spon

ding

med

ians

pro

port

iona

l to

corr

espo

ndin

g si

des.

vii.

Sim

ilar

tria

ngle

s ha

ve c

orre

spon

ding

ang

le b

isec

tors

pro

port

iona

l to

corr

espo

ndin

g si

des.

a.!R

S%

S ?T%

TR

!#

! URS V!

UV

%V

W%

WU

;iii

b.! UR

WT !#

!S ?X !V

Y;v

c.!R U

M N!#

! V? W!S

T;vi

d.! UR

S V!#

!S ?T !V

W;i

i

e.!R P

SP !#

! S? T!R

T;iv

f.!U ?N !

#!U R

W T!R

M;v

i

g.! WT

P Q!#

!R ?T !U

W;v

iih

.!U V

WW !#

! Q? V!U

Q;i

v

Hel

pin

g Y

ou

Rem

emb

er

2.A

goo

d w

ay t

o re

mem

ber

a la

rge

amou

nt o

f in

form

atio

n is

to

rem

embe

r ke

y w

ords

.Wha

tke

y w

ords

will

hel

p yo

u re

mem

ber

the

feat

ures

of

sim

ilar

tria

ngle

s th

at a

re p

ropo

rtio

nal

to t

he le

ngth

s of

the

cor

resp

ondi

ng s

ides

?pe

rim

eter

s,al

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es,a

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ecto

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edia

ns

W

QN

YU

V

T

PM

XR

S

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Pro

port

ions

for

Sim

ilar T

rian

gles

Rec

all t

hat

if a

line

cro

sses

tw

o si

des

of a

tri

angl

e an

d is

par

alle

l to

the

thir

dsi

de,t

hen

the

line

sepa

rate

s th

e tw

o si

des

that

it c

ross

es in

to s

egm

ents

of

prop

orti

onal

leng

ths.

You

can

wri

te m

any

prop

orti

ons

by id

enti

fyin

g si

mila

r tr

iang

les

in t

hefo

llow

ing

diag

ram

.In

the

diag

ram

,AM

! "#

& BN

!"# ,

AE

!"#

& FL

!"#

& MR

!"# ,

and

DQ

!"#

& ER

!"# .

An

swer

eac

h q

ues

tion

.Use

th

e d

iagr

am a

bove

.

1.N

ame

a tr

iang

le s

imila

r to

"G

NP

.2.

Nam

e a

tria

ngle

sim

ilar

to "

CJH

.

!A

MP

or !

GB

Aor

!A

FG!

CR

P

3.N

ame

two

tria

ngle

s si

mila

r to

"JK

S.

4.N

ame

a tr

iang

le s

imila

r to

"A

CP

.

Sam

ple

answ

er:!

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,!C

DS

!G

HP

Com

ple

te e

ach

pro

por

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5.!A A

G P!#

!A ?F !A

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! CCHP !

#!C ?R !

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7.! JJ RS !

#! J? L!

JK

8.!P PH C!

#!P ?G !

PA9.

!E LRR !

#! J? R!

CR

10.

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GA

Sol

ve.

11.I

f C

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nd L

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ind

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12.I

f D

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d C

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nd J

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o th

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14.I

f C

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82,a

nd C

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find

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15.W

rite

thr

ee m

ore

prob

lem

s th

at c

an b

e so

lved

usi

ng t

he d

iagr

am a

bove

.S

ee s

tude

nts’

wor

k.

AB

CD

E

FG

HJ

KL

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PR

Q

S

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-5

6-5



An

swer

s

Stu

dy

Gu

ide

and I

nte

rven

tion

Frac

tals

and

Sel

f-S

imila

rity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-6

6-6

©G

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5G

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eom

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Lesson 6-6

Ch

arac

teri

stic

s o

f Fr

acta

lsT

he a

ct o

f re

peat

ing

a pr

oces

s ov

er a

nd o

ver,

such

as

find

ing

a th

ird

of a

seg

men

t,th

en a

thi

rd o

f th

e ne

w s

egm

ent,

and

so o

n,is

cal

led

iter

atio

n.

Whe

n th

e pr

oces

s of

iter

atio

n is

app

lied

to s

ome

geom

etri

c fi

gure

s,th

e re

sult

s ar

e ca

lled

frac

tals

.For

obj

ects

suc

h as

fra

ctal

s,w

hen

a po

rtio

n of

the

obj

ect

has

the

sam

e sh

ape

orch

arac

teri

stic

s as

the

ent

ire

obje

ct,t

he o

bjec

t ca

n be

cal

led

self

-sim

ilar

.

In t

he

dia

gram

at

the

righ

t,n

otic

e th

at t

he

det

ails

at

each

sta

ge a

re s

imil

ar t

o th

e d

etai

ls a

t S

tage

1.

1.Fo

llow

the

iter

atio

n pr

oces

s be

low

to

prod

uce

a fr

acta

l.

Stag

e 1

• D

raw

a s

quar

e.•

Dra

w a

n is

osce

les

righ

t tr

iang

le o

n th

e to

p si

de o

f th

e sq

uare

.U

se t

he s

ide

of t

he s

quar

e as

the

hyp

oten

use

of t

he t

rian

gle.

• D

raw

a s

quar

e on

eac

h le

g of

the

rig

ht t

rian

gle.

Stag

e 2

Rep

eat

the

step

s in

Sta

ge 1

,dra

win

g an

isos

cele

s tr

iang

le a

nd t

wo

smal

l squ

ares

for

eac

h of

the

sm

all

squa

res

from

Sta

ge 1

.

Stag

e 3

Rep

eat

the

step

s in

Sta

ge 1

for

eac

h of

the

sm

alle

st s

quar

es in

Sta

ge 2

.

2.Is

the

fig

ure

prod

uced

in S

tage

3 s

elf-

sim

ilar?

yes

Stag

e 1

Stag

e 2

Stag

e 3

Exercis

esExercis

es

Exam

ple

Exam

ple

©G

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6G

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No

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c It

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ion

An

iter

ativ

e pr

oces

s ca

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app

lied

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n al

gebr

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expr

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on o

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on.T

he r

esul

t is

cal

led

a re

curs

ive

form

ula

.

Fin

d t

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valu

e of

x3 ,

wh

ere

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ial

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2.R

epea

t th

ep

roce

ss t

hre

e ti

mes

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d d

escr

ibe

the

pat

tern

.

Init

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st t

ime:

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The

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ach

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the

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d fo

r th

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xt s

tep.

For

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mpl

e,th

e x

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es a

re g

reat

er w

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iter

atio

n.T

here

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o m

axim

um v

alue

,so

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es a

rede

scri

bed

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ppro

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ng i

nfin

ity.

For

Exe

rcis

es 1

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ind

th

e va

lue

of e

ach

exp

ress

ion

.Th

en u

se t

hat

val

ue

as t

he

nex

t x

in t

he

exp

ress

ion

.Rep

eat

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pro

cess

th

ree

tim

es,a

nd

des

crib

e yo

ur

obse

rvat

ion

s.

1.$

x!,w

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itia

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qual

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e x

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ler

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2.!1 x! ,

whe

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!1 2! ;2,

!1 2! ,2;

the

xva

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,whe

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oach

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ity.

6.H

arpe

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aid

$100

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r a

savi

ngs

cert

ific

ate.

It e

arns

inte

rest

at

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l rat

e of

2.8

%,

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inte

rest

is a

dded

to

the

cert

ific

ate

each

yea

r.W

hat

will

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cer

tifi

cate

be

wor

th a

fter

four

yea

rs?

$111

6.79

Stu

dy

Gu

ide

and I

nte

rven

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tinu

ed)

Frac

tals

and

Sel

f-S

imila

rity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-6

6-6

Exercis

esExercis

es

Exam

ple

Exam

ple



Skil

ls P

ract

ice

Frac

tals

and

Sel

f-S

imila

rity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

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RIO

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___

6-6

6-6

©G

lenc

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w-H

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7G

lenc

oe G

eom

etry

Lesson 6-6

Sta

ges

1 an

d 2

of

a fr

acta

l k

now

n a

s th

e C

anto

r se

t ar

e sh

own

.To

get

Sta

ge 2

,th

ese

gmen

t in

Sta

ge 1

is

tris

ecte

d,a

nd

th

e in

teri

or o

f th

e m

idd

le s

egm

ent

is r

emov

ed.

(Th

e in

teri

or o

f a

segm

ent

is t

he

segm

ent

wit

h i

ts e

nd

poi

nts

rem

oved

.) T

he

pro

cess

is

rep

eate

d f

or s

ubs

equ

ent

stag

es.

1.D

raw

sta

ges

3 an

d 4

of t

he C

anto

r se

t.

2.H

ow m

any

segm

ents

are

the

re in

Sta

ge 3

? St

age

4?4

segm

ents

;8 s

egm

ents

3.W

hat

happ

ens

to t

he le

ngth

of

the

line

segm

ents

in e

ach

stag

e?Th

e lin

e se

gmen

ts d

ecre

ase

in le

ngth

by

two-

thir

ds a

t ea

ch s

tage

.

4.T

he C

anto

r se

t is

the

set

of

poin

ts a

fter

infi

nite

ly m

any

iter

atio

ns.I

s th

e C

anto

r se

t se

lf-s

imila

r?ye

s

Fin

d t

he

valu

e of

eac

h e

xpre

ssio

n.T

hen

,use

th

at v

alu

e as

th

e n

ext

xin

th

eex

pre

ssio

n.R

epea

t th

e p

roce

ss t

hre

e ti

mes

,an

d d

escr

ibe

you

r ob

serv

atio

ns.

5.2x

&3,

whe

re x

init

ially

equ

als

5

2(5)

#3

"7;

2(7)

#3

"11

;2(1

1) #

3 "

19;2

(19)

#3

"35

The

itera

tes

incr

ease

by

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ecut

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pow

ers

of 2

.

6.! 2x ! ,

whe

re x

init

ially

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als

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!1 2!"

0.5;

!0 2.5 !"

0.25

;!0.

225 !"

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0.06

25

The

itera

tes

are

halv

ed w

ith e

ach

itera

tion,

appr

oach

ing

zero

.

Fin

d t

he

firs

t th

ree

iter

ates

of

each

exp

ress

ion

.

7.x2

&1,

whe

re x

init

ially

equ

als

28.

!1 x! ,w

here

xin

itia

lly e

qual

s 8

3,8,

630.

125,

8,0.

125

Stag

e 3

Stag

e 4

Stag

e 1

Stag

e 2

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8G

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eom

etry

An

art

ist

is d

esig

nin

g a

book

cov

er a

nd

wan

ts t

o sh

ow a

cop

y of

th

e bo

ok c

over

in

the

low

er l

eft

corn

er o

f th

e co

ver.

Aft

er S

tage

s 1

and

2 o

f th

e d

esig

n,h

e re

aliz

esth

at t

he

des

ign

is

dev

elop

ing

into

a f

ract

al!

1.D

raw

Sta

ges

3 an

d 4

of t

he b

ook

cove

r fr

acta

l.

2.A

fter

infi

nite

ly m

any

iter

atio

ns,w

ill t

he r

esul

t be

a s

elf

sim

ilar

frac

tal?

yes

3.O

n w

hat

part

of

the

book

cov

er s

houl

d yo

u fo

cus

your

att

enti

on t

o be

sur

e yo

u ca

n fi

nd a

copy

of

the

enti

re f

igur

e?th

e lo

wer

left

cor

ner

Fin

d t

he

valu

e of

eac

h e

xpre

ssio

n.T

hen

,use

th

at v

alu

e as

th

e n

ext

xin

th

eex

pre

ssio

n.R

epea

t th

e p

roce

ss t

hre

e ti

mes

an

d d

escr

ibe

you

r ob

serv

atio

ns.

4.2(

x%

3),w

here

xin

itia

lly e

qual

s 0

5.! 2x !

&3,

whe

re x

init

ially

equ

als

10

6,18

,42,

902,

#2,

#4,

#5

The

diff

eren

ce b

etw

een

valu

es

The

diff

eren

ce b

etw

een

valu

es is

do

uble

s w

ith e

ach

term

.m

ultip

lied

by !1 2!

with

eac

h te

rm.

Fin

d t

he

firs

t th

ree

iter

ates

of

each

exp

ress

ion

.

6.! 2x !

%1,

whe

re x

init

ially

equ

als

17.

2x2 ,

whe

re x

init

ially

equ

als

2

1.5;

1.75

;1.8

758;

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68

8.H

OU

SIN

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he A

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ws

purc

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d a

hous

e fo

r $9

6,00

0.T

he r

eal e

stat

e ag

ent

who

sol

dth

e ho

use

said

tha

t co

mpa

rabl

e ho

uses

in t

he a

rea

appr

ecia

te a

t a

rate

of

4.5%

per

yea

r.If

thi

s pa

tter

n co

ntin

ues,

wha

t w

ill b

e th

e va

lue

of t

he h

ouse

in t

hree

yea

rs?

Rou

nd t

oth

e ne

ares

t w

hole

num

ber.

$109

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e 3 MAT

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age

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HM

ATH

Pra

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e (A

vera

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Frac

tals

and

Sel

f-S

imila

rity

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-6

6-6



An

swer

s

Rea

din

g t

o L

earn

Math

emati

csFr

acta

ls a

nd S

elf-

Sim

ilari

ty

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-6

6-6

©G

lenc

oe/M

cGra

w-H

ill32

9G

lenc

oe G

eom

etry

Lesson 6-6

Pre-

Act

ivit

yH

ow i

s m

ath

emat

ics

fou

nd

in

nat

ure

?

Rea

d th

e in

trod

ucti

on t

o L

esso

n 6-

6 at

the

top

of

page

325

in y

our

text

book

.

Nam

e tw

o ob

ject

s fr

om n

atur

e ot

her

than

bro

ccol

i in

whi

ch a

sm

all p

iece

rese

mbl

es t

he w

hole

.S

ampl

e an

swer

:rin

gs o

f a

tree

tru

nk;

spir

al s

hell

Rea

din

g t

he

Less

on

1.M

atch

eac

h de

fini

tion

fro

m t

he f

irst

col

umn

wit

h a

term

fro

m t

he s

econ

d co

lum

n.(S

ome

wor

ds o

f ph

rase

s in

the

sec

ond

colu

mn

may

be

used

mor

e th

an o

nce

or n

ot a

t al

l.)

Ph

rase

Term

a.a

geom

etri

c fi

gure

tha

t is

cre

ated

usi

ng it

erat

ion

iiii.

sim

ilar

b.a

patt

ern

in w

hich

sm

alle

r an

d sm

alle

r de

tails

of

a sh

ape

have

ii.i

tera

tion

th

e sa

me

geom

etri

c ch

arac

teri

stic

s as

the

ori

gina

l sha

peiv

iii.

frac

tal

c.th

e re

sult

of

tran

slat

ing

an it

erat

ive

proc

ess

into

a f

orm

ula

oriv

.se

lf-s

imila

ral

gebr

aic

equa

tion

vv.

recu

rsio

nd.

the

proc

ess

of r

epea

ting

the

sam

e pr

oced

ure

over

and

ove

rfo

rmul

aag

ain

iivi

.con

grue

nt

2.R

efer

to

the

thre

e pa

tter

ns b

elow

.

i. ii.

iii.

a.G

ive

the

spec

ial n

ame

for

each

of

the

frac

tals

you

obt

ain

by c

onti

nuin

g th

e pa

tter

nw

itho

ut e

nd.

i.K

och

snow

flake

;ii.

Sie

rpin

ski t

rian

gle;

iii.K

och

curv

eb.

Whi

ch o

f th

ese

frac

tals

are

sel

f-si

mila

r?S

ierp

insk

i tri

angl

e,K

och

curv

ec.

Whi

ch o

f th

ese

frac

tals

are

str

ictl

y se

lf-s

imila

r?S

ierp

insk

i tri

angl

e

Hel

pin

g Y

ou

Rem

emb

er3.

A g

ood

way

to

rem

embe

r a

new

mat

hem

atic

al t

erm

is t

o re

late

it t

o ev

eryd

ay E

nglis

hw

ords

.The

wor

d fr

acta

lis

rel

ated

to

frac

tion

and

frag

men

t.U

se y

our

dict

iona

ry t

o fi

ndat

leas

t on

e de

fini

tion

for

eac

h of

the

se w

ords

tha

t yo

u th

ink

is r

elat

ed t

o th

e m

eani

ng o

fth

e w

ord

frac

tal

and

expl

ain

the

conn

ecti

on.

Sam

ple

answ

er:F

ract

ion:

a sm

all

part

;a d

isco

nnec

ted

piec

e;fr

agm

ent:

a pi

ece

deta

ched

or

brok

en o

ff;

som

ethi

ng in

com

plet

e;in

a f

ract

al,t

he s

mal

ler

piec

es o

f th

e sh

ape

rese

mbl

e th

e w

hole

.

©G

lenc

oe/M

cGra

w-H

ill33

0G

lenc

oe G

eom

etry

The

Möb

ius

Str

ipA

Möb

ius

stri

p is

a s

peci

al s

urfa

ce w

ith

only

one

sid

e.It

was

dis

cove

red

byA

ugus

t Fe

rdin

and

Möb

ius,

a G

erm

an a

stro

nom

er a

nd m

athe

mat

icia

n.

1.To

mak

e a

Möb

ius

stri

p,cu

t a

stri

p of

pap

er a

bout

16

inch

es lo

ng a

nd 1

inch

wid

e.M

ark

the

ends

wit

h th

e le

tter

s A

,B,C

,and

Das

sho

wn

belo

w.

Tw

ist

the

pape

r on

ce,c

onne

ctin

g A

to D

and

Bto

C.T

ape

the

ends

toge

ther

on

both

sid

es.

See

stu

dent

s’w

ork.

2.U

se a

cra

yon

or p

enci

l to

shad

e on

e si

de o

f th

e pa

per.

Shad

e ar

ound

the

stri

p un

til y

ou g

et b

ack

to w

here

you

sta

rted

.Wha

t ha

ppen

s?

The

entir

e st

rip

is s

hade

d on

bot

h si

des.

3.W

hat

do y

ou t

hink

will

hap

pen

if y

ou c

ut t

he M

öbiu

s st

rip

dow

n th

em

iddl

e? T

ry it

.

Inst

ead

of t

wo

loop

s,yo

u ge

t on

e lo

op t

hat

is t

wic

e as

long

as t

he o

rigi

nal o

ne.

4.M

ake

anot

her

Möb

ius

stri

p.St

arti

ng a

thi

rd o

f th

e w

ay in

fro

m o

ne e

dge,

cut

arou

nd t

he s

trip

,sta

ying

alw

ays

the

sam

e di

stan

ce in

fro

m t

he e

dge.

Wha

t ha

ppen

s?

Two

inte

rloc

king

loop

s ar

e fo

rmed

.

5.St

art

wit

h an

othe

r lo

ng s

trip

of

pape

r.T

wis

t th

e pa

per

twic

e an

d co

nnec

tth

e en

ds.W

hat

happ

ens

whe

n yo

u cu

t do

wn

the

cent

er o

f th

is s

trip

?

Two

inte

rloc

king

loop

s ar

e fo

rmed

.

6.St

art

wit

h an

othe

r lo

ng s

trip

of p

aper

.Tw

ist

the

pape

r th

ree

tim

es a

ndco

nnec

t th

e en

ds.W

hat

happ

ens

whe

n yo

u cu

t do

wn

the

cent

er o

f thi

s st

rip?

One

tw

o-si

ded

loop

with

a k

not

is fo

rmed

.

A B

C D

En

rich

men

t

NA

ME

____

____

____

____

____

____

____

____

____

____

____

__D

ATE

____

____

____

PE

RIO

D__

___

6-6

6-6


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