Riverside Secondary 9.1 Scale Factor · 2020. 5. 21. · e) Square - A quadrilateral with four...

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Section 9.1 - Scale Factor ♦ 315

Polygon

Apolygonistheunionofthreeormoresegmentssuchthateachsegmentintersectsexactlytwoothers,oneat eachofitsendpoints(itsvertices).

Triangle

Athree-sidedpolygonwherethesumofitsinterioranglesaddto180°.

a) Scalene-Atrianglewithnosidesofthesamelength. b) Isosceles-Atrianglewithtwosidesofthesamelengthandtwoanglesequal. c) Equilateral-Atrianglewithallthreesidesequalandallthreeanglesequal.

IsoscelesTriangle ScaleneTriangleEquilateralTriangle

Polygon

Triangle Quadrilateral

Scalene

Trapezoid

Parallelogram

Rectangle Rhombus

Kite

Square

Isosceles Equilateral

Pentagon 5sides

Hexagon 6sides

Heptagon 7sides

Octagon 8sides

Nonagon 9sides

Decagon 10sides

Dodecagon 12sides

Scale Factor9.1

Riverside Secondary


316 ♦ Chapter 9 - Similarity and Scale Factors

Quadrilateral

Afoursidedpolygonwherethesumofinterioritsanglesis360°.

a) Trapezoid-Aquadrilateralwithonepairofparallelsides.

b) Parallelogram-Aquadrilateralwithtwopairsofparallelsides.

Bothpairsofoppositesidesareparallel. Bothpairsofoppositesidesareequal.

c) Rhombus-Aquadrilateralwithfourequalsides.

Therhombushasallthepropertiesofaparallelogram.

d) Rectangle-Aquadrilateralwithfourequalangles.

Therectanglehasallthepropertiesoftheparallelogram.

e) Square-Aquadrilateralwithfourequalsidesandfourequalangles.

Thesquarehasallthepropertiesofarhombusandarectangle.

f) Kite-Aquadrilateralwithtwodistinctpairsofconsecutivesidesofthesamelength.

Everykitehasatleasttwoanglesofequalmeasure.

h

b

a

h

b

a

h

b

h

b

Riverside Secondary



Similar Figures

Ifyouareaskedtomakeadrawingofapolygon,oranyfigure,youmightmakethedrawinglarger,smaller,or thesamesize.Butifyouwantthefigurestolookalike,theshapemustbethesame,andthecorresponding lengthshavetobeproportional.

Twofiguresaresimilarif:

a) Thecorrespondinganglesarethesame

b) Thecorrespondinglengthsareproportional

Wewritethesymbol+ toindicatethefiguresaresimilar.Forexample ABC DEFT T+

Scale Factor

Figuresthathavethesameshape,butdifferentsizes,arefoundthroughouteverydaylife.Maps,architects’ drawings,modelsofatoms,andmodeltrainsaresomeexamplesofsimilarfigures.Whenyoucomparea modeltraintotherealtrain,themodelismuchsmaller.Thisiscalledareduction.Whenyoucomparethe drawingofanatominyoursciencetexttotheactualatom,thedrawingismuchlarger.Thisiscalledan enlargement.

Scalefactor=object lengthimage length

a) Iftheimageislargerthantheobject,thescalefactorisgreaterthanone,anditisanenlargement.

b) Iftheimageissmallerthantheobject,thescalefactorisbetweenzeroandone,anditisareduction.

c) Iftheimageandobjectarethesame,thescalefactorisone.

Riverside Secondary



Considerthefollowingdiagram.Therearetwowaystodetermineitsscalefactor.

Method1: Determineacentrepointoftheobjectandimageasthedrawingshows.

MeasurePAPA

PBPB

PCPC

PDPD

= = = =l l l l scalefactor

Method2: MeasureABA B

BCB C

CDC D

DAD A

= = = =l l l l l l l l scalefactor

Alloftheseratioswillgivethesamescalefactor.

Determinethescalefactoroftheabovedrawing.

►Solution: Method1:Scalefactor=.. 2.42

PCPC

cmcm

3 58 5

= =l

Method2:Scalefactor=.. .

ABA B

cmcm

1 02 4 2 40= =

l l

Noticethatthescalefactorsfromthetwomethodsareslightlydifferent.Thisisdueto inaccuracyofmeasurements.

C’

D’

C

DA’

object imageP

B

A

B’

(centre)

Example 1

Riverside Secondary



Giventhelengthsandanglesshowninthediagram

a) Whyis ABCD EFGH4 4+ ? b) Determinethescalefactor. c) If 4AD = cm,determineEF.

►Solution: a) Interioranglesof4ABCDaddto360°,so+Ais105° Interioranglesof4EFGHaddto360°,so+His80°

Therefore,allthecorrespondinganglesofthequadrilateralareequal.

b) Scalefactor object sizeimage size

= ADEF

= ..3 83 0= .0 8=

c) 0.8 . .EF AD 4 0 8 3 2# #= = = cm

Determinethescalefactorofthediagrambelow.

►Solution: Scalefactor .105

21 0 5object size

image size= = = =

Example 2

Example 3

y

x

B’(4,2)

A’(5,4)

C’(7,3)

B(8,4)

A(10,8)

C(14,6)

0

85°80°

85°

105°

A

BC

DE

GH

F

Object Image

Riverside Secondary



9.1 Exercise Set

1. Dotheobjectshavethesameshapeasafootballfieldmeasuring160feetby360feet?

a) Asheetofpapermeasuring5cmby10cm. y/n

b) Arectanglemeasuring4cmby9cm. y/n

c) Aparallelogramwithsides16by36. y/n

d) Aphotographmeasuring6cmby13.5cm. y/n

e) Amagazinemeasuring9cmby22.5cm. y/n

2. Completethetableforeachenlargement.

ObjectLength ImageLength ScaleFactor12cm 24cm30cm 40cm9cm 4

48cm 1.550cm 2.5

3. Completethetableforeachreduction.

ObjectLength ImageLength ScaleFactor25cm 5cm

10cm 0.165cm 0.214.7cm 0.6

5cm 0.4

4. Usearulertodeterminethecentreandthescalefactorofthefollowing:

a) b)

Thescalefactoris. Thescalefactoris.

image

object

object

image

Riverside Secondary



5. Drawtheimageofthebaseballifthescalefactor 6. Drawtheimageofthefootballifthescalefactor is 2

1 . is1.5.

7. Aportraitmeasures50cmby35cm.Ifthelarger 8. Adollhouseisonetenthoftheactualsizeofahouse. sideofareductionmeasures20cm,whatisthe Ifachairinthedollhouseis2.75cm,howtallisthe lengthoftheshortersideofthereduction? actualchairitmodels?

9. Apole3mhighhasashadow5mlong.Ifa 10.A193kglunarvehicleweighs31kgonthemoon. buildingis66mhigh,howlongisthebuilding’s Howmuchdoyouweighonthemoon? shadow?

11. Whatisthescaleofthesoccerfield?Giveanswer 12.Ahockeyrinkis96mby48m.Drawascalemodel as1cm=yds. ofthehockeyrinkifthescaleis1cmto12m.

120yds

75yds

Riverside Secondary



13.

a) IfthescaleofthemapofCanadais b) IfthescaleofthemapofCanadais 1cm=460km,whatistheactualdistance 1cm=460km,howmuchfartherinkilometers betweenCalgaryandHalifax? isitbetweenCalgaryandHalifaxthanbetween VancouverandWinnipeg?

c) Ifitis2600kmbetweenSaskatoonand d) IfthedistancebetweenFortMcMurrayand Montreal,howfarisitbetweenVancouver Winnipegis1200km,whatisthescaleofthe andToronto? mapofCanada?

e) IfthescaleofCanadais1cm=460km, f) IfthescaleofthemapofCanadais1cm=460km, howlongwouldaplanetravellingat howmuchlongerwouldaplanetravellingat 600km/htaketogetfromCalgaryto 750km/htaketoflyfromCalgarytoHalifaxthan Halifax? fromVancouvertoWinnipeg?

.Saskatoon

.Montreal

. Toronto

Vancouver..Halifax

Fort McMurray.

.Winnipeg.

Calgary

Riverside Secondary



14. Draweachimagemagnifiedbyascalefactorof2andby21 .

a) b)

15. Findthescalefactor.

a) b)

ScaleFactor ScaleFactor

16. Grapheachimageasindicated.

a) Scalefactor3 b) Scalefactor21 .

0

object

image

object

image

0

00

0 0

Riverside Secondary



Twopolygonsaresimilarif:

a)Thecorrespondinganglesareequal. b)Thecorrespondingsidesareproportional.

Iftwoanglesofonetriangleequaltwoanglesofanothertriangle,thenthetrianglesaresimilar(~).

If A D+ += and C F+ += then B E+ += so ABC DEFT T+

(If two angles are equal, the third angle is also equal because the three interior angles of a triangle add to 180°)

Similartriangleshavecorrespondingsidesthatareproportional.

If ABC DEFT T+ ,thenDEAB

DFAC

EFBC

= = .

Similarity Properties in Right Triangles

Thealtitudetothehypotenuseofarighttriangleformstwotrianglesthataresimilartoeachotherandtothe originaltriangle.

ABC BDC ABDT T T+ + hc

dh

ba

= = , ae

ca

hb

= = , be

db

ha

= =

E

FA

B

C

~D

A

B

D

a

b

c

d

e

h

C

Similar Triangles9.2

Riverside Secondary


Section 9.2 - Similar Triangles ♦ 325

Thesymbol~isusedforthephrase“issimilarto”.

If A D+ += , B E+ += and C F+ += thenthecorrespondinganglesareequal,thereforethetwotrianglesare similar,writtenas ABC DEFT T+ .Sincethetrianglesaresimilar,thecorrespondingsidesareproportional.

Namepairsofequalanglesandequalratiosofsidesinthetwosimilartriangles.

►Solution: Equalanglesare A D+ += , B E+ += , C F+ +=

Ratiosofsidesareda

eb

fc

= =

Determinethelengthofxandy.

►Solution: x6

2 5=

y62 7=

x2 30= y2 42=

x22

230

= y22

242

=

x 15= y 21=

Example 1

Example 2

E

FDA

B

a

b

cdf

eC

E

FA

B

C

~D

~2 5

7

6 x

y

Riverside Secondary



Namepairsofequalanglesandequalratiosofsidesinthetwosimilartriangles.

►Solution: Equalanglesare ABF C+ += , AFB D+ += , A A+ += .

RatiosofsidesareACAB

ADAF

CDBF

= = .

Calculatethevalueofx.

►Solution: x3 64 6

= +

x3 610

=

x6 3 10#=

x66

630

=

x 5=

Calculatethevalueofx.

►Solution: Since E A+ += ,and C C+ += ,then B D+ += .Therefore ABC EDCT T+ .

EDAB

DCBC=

x412

36= +

x4 24 36+ =

4 12x =

x 3=

Example 3

Example 4

Example 5

F

A

B

C D

4

3

6

x

E

A

B C

124 3

6x

D

Riverside Secondary



Calculatethevalueofy.

►Solution: ABC DECT T+

DEAB

CDAC

CEBC

= =

y5106

=

y6 5 10#=

y66

650

=

y325

=

Calculatethevalueofz.

►Solution: Therighttrianglesaresimilar,thereforetheequalanglesare A BCD+ += , B ACD+ += , ACB ADC+ += .

DCAD

DBDC

=

z

z188

=

z 18 82 #=

z 18 8#=

z 12=

Example 6

Example 7

E

A

B

CD

510

6 y

D

A

B

C

18

8

z

Riverside Secondary



9.2 Exercise Set

1. Whichproportionsareequivalentto x8 32

= ?

a) x3 28

= b) x2 38

= c) x8835+ = d)

x8

32=

2. Completeeachstatement.

a) Ifyx52

= ,then5x = b) Ifba73

= ,then a3=

c) Ifnm

59

= ,thenmn= d) If

zy114

= ,thenzy z+

=

e) If c d4 3= ,then

dc= f) If

yx

411

= ,thenx11=

g) If a b5 12= ,then a

55+ = h) If

yx y

73-

= ,thenyx=

3. Findthevalueofx.

a) x7 43

= b) x4329+ =

c)x975

= d) x x34

58+ = +

e) x x32

74- = + f) x x

54

73- = -

Riverside Secondary



4. Nametheratioofsidesforthesimilartriangles.

a) b)

c) d)

e) f)

g) h)

A

B

D

E

C

F

ab

c

d

f

e

ab

cd

f

e ab

c d

f

e

a

c

f

e

b

d

ac

e

b

d

E

A

B

CD

E

A

B

C

D

Riverside Secondary



5. Foreachpairoftrianglesdeterminewhetherthetwotrianglesaresimilar.

a) b)

c) d)

e) f)

g) h)

10

6 8

5

3 4

1218

8

1 41

65

1 32

2 21

8

8

15

5

2524

7

12 13

5

75˚

65˚40˚

40˚

5

22

5

Riverside Secondary



6. Solveforthemissingside,orsides.

a) b)

x= x=

y= y=

c) d)

x=

y= x=

e) f)

x= x=

g) h)

x= x=

y= y=

65

10

12

xy7

9

912

x

y

7

10

8

16x

y

5

2 3

x4

5

2

x

9

4

x

y

x3 4

62

3 4

46 x

y

E

A

BC

D

Riverside Secondary



7. Atreecastsa5metershadowatatimewhena 8. Ann,whois155cmtall,placesamirroronthe flagpoleof8meterscastsashadowof1.5meters. ground115cmfromherfeet.Themirroris620cm Whatistheheightofthetree? fromaflagpole.IfAnn,bylookingatthemirror, seesthetopoftheflagpole,howtallistheflagpole?

9. Theslopetriangleonatruss 10. Alltelevisionscreenshavedimensionsthatare tellsyouhowhighthetruss proportional.Ifa50inchtelevisionscreen iscomparedtotherun. (measuredalongthediagonal)is43.5incheswide, Whatistheheightof howwideisa36inchscreen? thistrussgivena 12

5 slopeangle?

11. Findx . 12. Findx andy .

13. Thelengthofthesidesofoneoftwosimilartriangles 14. Inordertobesafe,a6mladdershouldbe2.5m area,b,andc witha b c1 1 .Ifthelongestsideof fromawall.Howfarfromawallshoulda8m theothertriangleisx,whatisthelengthofthetwo ladderbeplacedinordertobesafe? othersides?

10m

h

x

5 813

x

y

28

2420

20

Riverside Secondary



15. Acard12cmlongisfoldedinhalfalongitslength. 16. Acard12cmlongisfoldedalongitslengthinto Ifthefoldedcardhasthesameshapeastheoriginal fourrectangleswithdimensionsproportionaltothe card,whatisthewidthofthecardbeforefolding? original.Whatisthewidthoftheoriginalrectangle?

17. Toestimatethewidthofariver,asurveyormadea 18. Howmanysimilartrianglesarethere? diagramasshown.Whatisthewidthoftheriver?

19. Inthediagram,l,m,andnareparallel.

a) Ife 15= ,f 24= ,andg+ h 52= ,whatis b) Ifg 12= ,h 20= ,andf 2= e- 3,whatisthe thevalueofg? valueofe?

c) Ifg 16= ,h 24= ,e x50= - ,andf x= , d) Ifg 12= ,h 18= ,e x8= + ,andf x10= + , whatisthevalueofx? whatisthevalueofx?

8m

12m40m

l

m

n

e

f

g

h

Riverside Secondary



PolygonABCDEandpolygonFGHIJaresimilarpolygons

Therefore A F+ += , B G+ += , C H+ += , D I+ += , E J+ += andFGAB

GHBC

HICD

IJDE

JFEA

= = = =

Given ABCD EFGD4 4+ withAB=10cm,AD=6cmandAE=2cm,determineEF.

►Solution: IfAD=6cmandAE=2cm,thenED=4cm

EFAB

EDAD

=

EF10

46

=

6EF 4 10#=

EF320 6

32

= = cm

Example 1

A B

CD

E F

G

A

BC

DE

F

G

H

I

J

Similar Polygons9.3

Riverside Secondary


Section 9.3 - Similar Polygons ♦ 335

Given ABCD EFGH4 4+

Find a) E+ b) G+ if C 100c+ = c) EF d) CD e) Thescalefactorofimage ABCD4 toobject EFGH4 f) Thescalefactorofimage EFGH4 toobject ABCD4

►Solution: a) E 90c+ = Correspondinganglesareequal

b) G 100c+ = Correspondinganglesareequal

c)EFAB

EHAD

= Correspondingsidesareproportional

EF2

123

=

3 EF 2 12# #=

33EF

324

=

EF 8=

d)EHAD

GHCD

=

123

10CD

=

12 CD 3 10# #=

1212CD

1302

=

CD25

=

e) ScalefactorEHAD

123

41

= = = whichisareduction.

ABCD4 is41 thesizeof EFGH4

f) ScalefactorADEH

312

14

= = = whichisanenlargement.

EFGH4 is4timesthesizeof ABCD4

Example 2

B

C

D3

2

A E

F

G

H12

10

Riverside Secondary



Given ABCD EFGH4 4+

Determinea)Thescalefactorb)x,yandz

►Solution: a) Scalefactorobject sizeimage size

3220

4 8

4 585 0.625

#

#= = = = =

b) x2032

25=

y2032 36=

z2032 24=

x20 32 25#= y20

32 36= z32 20 24#=

x20

32 25#= y32

20 36#= z32

20 24#=

x 40= .y 22 5= z 15=

Tworectangleshavethesameshape,butdifferentareas.Iftheareashaveascalefactorof 9to16,whatisthescalefactorofthesidesoftherectangles?

►Solution:

ll

AA

ll

ll

169

16

943

2

12

2

1

22

12

2

1

2

" "= = = =

Thescalefactorofthesidesis3to4.

Tworectangularsolidshavethesameshape,butdifferentvolumes.Ifthevolumeshaveascale factorof8to27,whatisthescalefactorofthesidesoftherectangularsolids?

►Solution:ll

VV

ll

ll

27

8278

32

23

13

2

1

23

13

2

1

3

3

" "= = = =

Thescalefactoris2to3.

A

B

C

D

36

32

24

x

Object

E

FG

H

20

25

y z

Image

Example 3

Example 4

w

l A = 9

w

l A=16

Example 5

Riverside Secondary



Constructing Similar Polygons

Toconstructasimilarpolygonfromagivenpolygon,itisimportanttoremember: a) Allcorrespondinganglesmustbeequal. b) Allcorrespondingsidesmustbeproportional. c) Thescalefactorthatisneeded.

a) DrawpolygonA B C D El l l l l~polygonABCDEwithscalefactorof polygonABCDEtopolygonA B C D El l l l l=2.

b) Determineallcorrespondinganglesandratiosofcorrespondingsides.

CorrespondingAngles CorrespondingSidesA+ A+ l 143c AB=3 A B 1.5=l l

A BAB 2=l l

B+ B+ l 90c BC=4 B C 2=l lB CBC 2=l l

C+ C+ l 127c CD=4 C D 2=l lC DCD 2=l l

D+ D+ l 90c DE=5 D E 2.5=l lD EDE 2=l l

E+ E+ l 90c EA=4 E A 2=l lE AEA 2=l l

►Solution:

Example 6A

B

C

DE

3

4

4

5

4

53˚

Riverside Secondary



9.3 Exercise Set

1. Tellwhetherthetwopolygonsaresimilar:always,sometimesornever.

a) Twosquares

b) Tworectangles

c) Twoequilateraltriangles

d) Twoisoscelestriangles

e) Tworegularsix-sidedpolygons

f) Twopentagons

g) Ascalenetriangleandanisoscelestriangle

h) Tworhombuseswithequalangles

i) Twobasketballcourts

j) Tworegulationsizedtenniscourts

k) Anequilateraltriangleandarighttriangle

2. Classifythefollowingassimilar,notsimilar,orpossiblysimilar,tothegivendiagram.

a) b)

c) d)

50˚

130˚

20

13

16

25

50˚

130˚

50˚

130˚

50˚130˚

5

64

3

50˚

130˚

10

6

8

12

Riverside Secondary



3. Aquadrilateralhassidesa,b,c,d.Asecond 4. Thedimensionsofarectanglearex cmbyy cm. quadrilateralhassides2a,2b,2c,2d.Arethesides Asecondrectanglehasdimensionsx 4+ cmby similar?Why? y 4+ cm.Aretherectanglessimilar?Why?

5. Thedimensionsofarectanglearex cmbyy cm. 6. Thesidesofapentagonare2,3,2,4,5cm.Find Asecondrectanglehasdimensions x3 cmby y3 cm. theperimeterofasimilarpentagonwithtwosides Arethesidessimilar?Why? thatare3cm.

7. Determinethescalefactorofthefollowingsimilarpolygons,withtheobjectontheleftanditsimageonright. Taketheaverageofthreecalculations.Oneofthemeasurementsmustbewithoutmeasuringthelengthofsides ofeachpolygon.

a)

ScaleFactor=

b)

ScaleFactor=

c)

ScaleFactor=

d)

ScaleFactor=

Riverside Secondary



8. Thetwopolygonsineachproblemaresimilar.Findthemissingmeasures.

a)

x =

y =

z =

b) A+ = D+ =

E+ = H+ =

x = y =

z =

c) a =

b =

c =

d =

d) Findasmanymissinglengthsand anglesaspossible.

e) Findasmanymissinglengthsand anglesaspossible.

1525

3

20

x

y

z7

27

14

12

b

d10

12

6

a

c

45

30

P

Q

N M

O

4

B

C

DA 7

x

y

60˚

120˚408

E

F

G

Hz

32

40˚115

150

B

CD

A

E

120˚

130˚30˚

30B

C

D

A E

F

X

W

U

V

Z

Y

Riverside Secondary



9. Ifeachdimensionofarectangleisincreasedby 10. Ifeachdimensionofarectangleisincreasedby 25%,isthenewrectanglesimilartotheoriginal? 25%,whatisthescalefactoroftherectangles?

11. Theperimeterofahexagonis20cmandthelongest 12. ADEFisaparallelogram sideis5cm.Whatisthelongestsideofasimilar inscribedinsideT ABC. hexagonwhoseperimeteris32cm? Whatdoesequal?

13. Thescalefactoroftheareasoftwocirclesis25to 14. Thescalefactorofthevolumeoftwocubesis8to 36.Whatisthescalefactoroftheircircumferences? 27.Whatisthescalefactorofthesidesofthecubes?

15. Thevolumesoftwospheresare288r and7776r . 16. Thevolumesoftwospheresare288r and7776r . Whatistheratiooftheirradii? Whatistheratiooftheirsurfaceareas?

EF FAAD DE$$

A

BC

D

E

F

Riverside Secondary



Wehaveallseenourselvesinamirrororourreflectioninalake.Itispossibletotraceadrawingbyfolding apieceofpaperandtracingitsoutline.Theseareexamplesofsymmetryandlinereflection.

Thedottedlinemakeseachhalfofeachfigurethereflectionoftheotherhalf.Thesedottedlinesarelines of symmetry.

Tesselation

Acoveringofaplanewithcongruentcopiesofthesameregion,withnoholesandnooverlaps.

Example:

9.4

Riverside Secondary


Section 9.4 - Reflection and Symmetry ♦ 343

Reflection

AreflectionmapseverypointPtoapointPlsuchthat

a) IfPisnotonthereflectingline,thenthelineistheperpendicularbisectorofPPl b) IfPisontheline,thenP P= l

Inotherwords,areflectionisatransformationthatflipsafigureorobjectoveraline.

AtrianglehasverticesA(2,2),B(3,−3),C(−4,1)

a) Graph ABCT anditsimage A B CT l l lwhere A B CT l l lis ABCT reflectedoverthey-axis.

b) Describetheimageasamappingofageneralpoint(x,y).

c) Graph ABCT andimage A B CT l l lwhere A B CT l l lis ABCT reflectedoverthex-axis.

d) Describetheimageasamappingofageneralpoint(x,y).

►Solution: a) b) , ,x y x y" -^ ^h h

c) d) , ,x y x y" -^ ^h h

Example 1

x

y

A

B

CA´

B´

C´

x

y

A

B

C

A´

B´

C´

reflecting line

imageobject

Riverside Secondary



9.4 Exercise Set

1. Determinethelinesofsymmetry(ifpossible).

2. Completeeachshapesothatthedottedlineisalineofsymmetry.

a) b) c)

d) e) f)

g) h) i)

Riverside Secondary



3. Determinethenumberoflinesofsymmetryforthegeometricfigures.

a) Anequilateraltriangle b) Asquare

c) Aregularpentagon d) Aregularhexagon

e) Aregularheptagon f) Aregularoctagon

4. Drawthereflectinglineinthediagram.

a) b)

c) d)

Riverside Secondary



5. Sketchtheimageofeachfigureusingthegivenreflectingline.

a) b)

c) d)

6. Makeaholeinonebypropertyofreflection.

a) Drawalineindicatingaholeinone byrollingtheballoffwall“a”.

b) Drawalineindicatingaholeinoneby rollingtheballofwalls“b”then“a”.

7. Toseeyourfullreflectioninamirror,indicatethefollowing:

a) Wheremustthetopofthemirrorbeplacedonthewallsothatyoucanseethetopofyourhead?

b) Whatistheminimumheightofthemirror?

c) Wheredoyouseeyourselfinthemirror?(whereisyourmirrorimage)

hole

wall “a”

wall “b”

ball

P

Gwall

Riverside Secondary



8. Inbilliards,youare“snookered”whenthecueballcannothittheobjectballdirectly.

a) Indicatethepathofthecueball tohittheeightballwithone bankoffcushion“b”.

b) Indicatethepathofthecueball tohittheeightballwithabank offcushion“a”then“b”.

9. ThetrianglewithverticesA(2,2),B(3,−3), andC(−4,1)isreflectedoverthex-axis.

a) Graph ABCT anditsimage A B CT l l l.

b) Whatistheimagemappingoverthe x-axis? , ,x y "^ ^h h

10. ThetrianglewithverticesA(1,3),B(−2,3), andC(3,5)isreflectedovertheliney x= .

a) Graph ABCT anditsimage A B CT l l l.

b) Whatistheimagemappingoverthe liney x= ? , ,x y "^ ^h h

cue ball

eight ball

C U S H I O N “a”

C U S H I O N “b”

y

x

y

x

Riverside Secondary



11. Ifthepoint(1,3)isreflectedoverthegivenline,whatistheimagepoint?

a) x-axis

b) y-axis

c) y x=

d) x 5=

e) y 4=

12. Ifthepoint ,4 2-^ hisreflectedoverthegivenline,whatistheimagepoint?

a) x-axis

b) y-axis

c) y x=

d) x 1=

e) y 1=-

13. WhenthewordTOTisreflectedonaverticalline, 14. WhenthewordHIDEisreflectedonahorizontal theimageisunchanged.Thinkofthreewordsthat line,theimageisunchanged.Thinkofthreewords areunchangedwhenreflectedonaverticalline. thatareunchangedwhenreflectedonahorizontal line.

y

x

y

x

Riverside Secondary



15. Shadeintheunitwhichrepeatstomakethistesselation.Whatfractionoftheareaofthetesselationismade of:smallsquares,largesquares,andrectangles?

a)

Smallsquares

Largesquares

b)

Smallsquares

Rectangles

Largesquares

c)

Smallsquares

Rectangles

Largesquares

Riverside Secondary



16. Drawthreetesselationsofyourown.

a)

b)

c)

Riverside Secondary


Section 9.5 - Rotation ♦ 351

Inaturningmotion,allpointsintheoriginalfigureturnthesamenumberofdegreesaboutafixedcentrepoint. Thecentreofrotation,withadirectionandanamountdefinearotation.

Counterclockwiseisthepositivedirection.

Clockwiseisthenegativedirection.

Counterclockwise+60° clockwise−90°

Rotatethefigureinthediagram90°aboutpointA

►Solution: 90°ispositive,thereforetherotationiscounterclockwise.

Example 1

figure

image

centre point .

|

|| |

| |

| | |

| | | .A

B

C

A’

B’C’

y

x0clockwise -90

Rotation9.5

Riverside Secondary



Centre of Rotation

Tofindthecentreofrotationoftwoobjects,wedrawtheperpendicularbisectorofatleasttwocorresponding points.Thecentreofrotationiswherethetwoperpendicularbisectinglinesmeet.

Locatethecentreofrotation.

►Solution:

Thecentreofrotationis ,21 1-` j.

Locatethecentreofrotation.

►Solution:

Thecentreofrotationis ,1 1^ h.

Example 2

Example 3

Riverside Secondary



9.5 Exercise Set

1. a) Rotatethefigure180°aboutthecentre b) Rotatethefigure180°aboutthecentre ofrotation,pointP. ofrotation,pointP.

c) Rotatethefigure90°aboutthecentre d) Rotatethefigure−90°aboutthecentre ofrotation,pointA. ofrotation,pointA.

2. Rotatetheimageaboutthepointindicated,andlabelvertices.

a) Rotate41 turncounterclockwise. b) Rotate

41 turnclockwise.

.P

.A

.

y

x .

y

x

.P

.A

Riverside Secondary



3. TrianglesI,IIandIIIarerotationimagesof ABCT .Findthecentreofeachrotation.

TriangleI

TriangleII

TriangleIII

4. Rotatingthehexagon60degreesaroundpointOiswrittenasR0,60 .Determinethepointthatreplaces“A”after thegivenrotationisperformed.

Example: RO,60 → B

a) RO, 60- →

b) RO,300 →

c) RO, 120- →

d) RO,540 →

5. Nametwodifferentrotationsthatwillallowaferriswheelridertoexittheferriswheel.

Example: RiderA: RO,30 ,RO, 330-

a) RiderB: ,

b) RiderD: ,

c) RiderH: ,

d) RiderI: ,

6. Locatethecentreofrotation.

a) b)

A

B

CI II

III

.O

A

B

C

D

E

F

A

B

C

D

E F

G

H

I

JK

EXIT

figure

image

figure

image

Riverside Secondary




a) b)

8. Ahalf-turnaboutthepoint(2,2)maps(2,−1)to(2,5).Wheredoesahalfturnmapthefollowingpoints about(2,2).

a) (0,0)

b) (3,0)

c) (0,3)

d) (−4,−1)

9. Theverticesof ABCT areA(2,1),B(6,1),andC(5,3).Thetriangleisrotated180°abouttheorigin. Graphthetriangleanditsimage.Labelthevertices.

10. Theverticesof ABCT areA(−6,−1),B(−5,−4),andC(−1,−4).Thetriangleisrotated90°abouttheorigin. Graphthetriangleanditsimage.Labelthevertices.

y

A

BC B’

A’

C’

x

y

B

C

D

A

A’B’

C’ D’

x

y

. (2,2)

x

y

x

y

x

Riverside Secondary



Section 9.1

1. Completethetable.

ObjectLength ImageLength ScaleFactor8m 6m15cm 25cm12in 1.6

12in 1.624mm 0.8

24mm 0.8

2. Determinethescalefactorofthefollowing.

a) b)

Object Image Object Image

Thescalefactoris. Thescalefactoris.

Section 9.2

3. Completeeachstatement.

a) Ifba b

57+ = ,then

ba = . b) If

ba b

32- = ,then

ba = .

c) Ifba

31= ,then

ba b+ = . d) If

ba

56= ,then

ba b- = .

Chapter Review9.6

Riverside Secondary


Section 9.6 - Chapter Review ♦ 357

4. Findthevalueofx.

a) x32

73+ = b) x x

42

83- = +

5. Solveforthemissingside.

a) b)

x=

y= x=

Section 9.3

6. Findthemissingmeasuresofthesimilarpolygons.

a=

b=

c=

d=

7. Ifitcosts$200tostainadeckmeasuring4mby 8. Apaintingmeasuring120cmwideby80cmtall 5m,whatwoulditcosttostainadeck5mby7m? ismountedinapictureframe.Thereisamargin of6cmwideatthetopandeachsideofthe painting.Ifthepaintingissimilartotheframe, whatmustbethewidthofthemarginatthe bottomofthepainting?

8

3

5 9

x

y

18

8

12

b

d 6

9

10

a

c

3

2

x

Riverside Secondary



Section 9.4

9. Completethefollowingstatements.

a) Whenanobjectisreflectedoverthex-axis,(x,y)→.

b) Whenanobjectisreflectedoverthey-axis,(x,y)→.

c) Whenanobjectisreflectedovertheliney=x,(x,y)→.

Section 9.5

10. Counterclockwiseisthepositive/negativedirection.(circleone)

Clockwiseisthepositive/negativedirection.(circleone)

11. RotatethefigureaboutpointO.

a) 90° b) −90°


a) b)

O O

Riverside Secondary

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