Transformations & Symmetry

The Mathematics Vision Project Scott Hendrickson, Joleigh Honey, Barbara Kuehl, Travis Lemon, Janet Sutorius

© 2016 Mathematics Vision Project Original work © 2013 in partnership with the Utah State Off ice of Education

This work is licensed under the Creative Commons Attribution CC BY 4.0

WCPSS Math 2 Unit 1: MVP MODULE 6

Transformations & Symmetry

SECONDARY

MATH ONE

An Integrated Approach

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SECONDARY MATH 1 // MODULE 6

TRANSFORMATIONS AND SYMMETRY

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org

MODULE 6 - TABLE OF CONTENTS

TRANSFORMATIONS AND SYMMETRY

6.1 Leaping Lizards! – A Develop Understanding Task

Developing the definitions of the rigid-motion transformations: translations, reflections and

rotations (NC.M2.G-CO.4, NC.M2.G-CO.5, NC.M2.F-IF.1, NC.M2.F-IF.2)

READY, SET, GO Homework: Transformations and Symmetry 6.1

6.3 Leap Frog – A Solidify Understanding Task

Determining which rigid-motion transformations will carry one image onto another congruent

image (NC.M2.G.CO.4, NC.M2.G.CO.5, NC.M2.F-IF.1, NC.M2.F-IF.2)


6.4 Leap Year – A Practice Understanding Task

Writing and applying formal definitions of the rigid-motion transformations: translations, reflections

and rotations (NC.M2.G.CO.2, NC.M2.G.CO.4)


6.5 Symmetries of Quadrilaterals – A Develop Understanding Task

Finding rotational symmetry and lines of symmetry in special types of quadrilaterals (NC.M2.G.CO.3)


6.6 Symmetries of Regular Polygons – A Solidify Understanding Task

Examining characteristics of regular polygons that emerge from rotational symmetry and lines

of symmetry (NC.M2.G.CO.3)


6.7 Quadrilaterals—Beyond Definition – A Practice Understanding Task

Making and justifying properties of quadrilaterals using symmetry

transformations (NC.M2.G.CO.3, NC.M2.G.CO.4)


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SECONDARY MATH I // MODULE 6

TRANSFORMATION AND SYMMETRY – 6.1




6. 1 Leaping Lizards!

A Develop Understanding Task

Animatedfilmsandcartoonsarenowusuallyproducedusingcomputertechnology,

ratherthanthehand-drawnimagesofthepast.Computeranimationrequiresbothartistic

talentandmathematicalknowledge.

Sometimesanimatorswanttomoveanimagearoundthecomputerscreenwithout

distortingthesizeandshapeoftheimageinanyway.Thisisdoneusinggeometric

transformationssuchastranslations(slides),reflections(flips),androtations(turns),or

perhapssomecombinationofthese.Thesetransformationsneedtobepreciselydefined,so

thereisnodoubtaboutwherethefinalimagewillenduponthescreen.

Sowheredoyouthinkthelizardshownonthegridonthefollowingpagewillendup

usingthefollowingtransformations?(Theoriginallizardwascreatedbyplottingthe

followinganchorpointsonthecoordinategrid,andthenlettingacomputerprogramdrawthe

lizard.Theanchorpointsarealwayslistedinthisorder:tipofnose,centerofleftfrontfoot,

belly,centerofleftrearfoot,pointoftail,centerofrearrightfoot,back,centeroffrontright

foot.)

Originallizardanchorpoints:

{(12,12),(15,12),(17,12),(19,10),(19,14),(20,13),(17,15),(14,16)}

Eachstatementbelowdescribesatransformationoftheoriginallizard.Dothe

followingforeachofthestatements:

• plottheanchorpointsforthelizardinitsnewlocation

• connectthepre-imageandimageanchorpointswithlinesegments,orcirculararcs,

whicheverbestillustratestherelationshipbetweenthem

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LazyLizard

Translatetheoriginallizardsothepointatthetipofitsnoseislocatedat(24,20),makingthe

lizardappearstobesunbathingontherock.

LungingLizard

Rotatethelizard90°aboutpointA(12,7)soitlookslikethelizardisdivingintothepuddle

ofmud.

LeapingLizard

Reflectthelizardaboutgivenline soitlookslikethelizardisdoingabackflip

overthecactus.

€

y = 12 x +16

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TRANSFORMATIONS AND SYMMETRY - 6.1




6.1

READY

Topic:PythagoreanTheorem

Foreachofthefollowingrighttrianglesdeterminethemeasureofthemissingside.Leavethemeasuresinexactformifirrational.

1. 2. 3.

4. 5. 6.

READY, SET, GO! Name Period Date

3

4

?

?

5

12

?4

1

?

3

√10

?

√174

?

2

√13

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6.1

SET Topic:Transformations.

Transformpointsasindicatedineachexercisebelow.

7a.RotatepointAaroundtheorigin90oclockwise,labelasA’

b. ReflectpointAoverx-axis,labelasA’’

c. Applytherule(! − 2 , ! − 5),topointAandlabelA’’’

8a.ReflectpointBovertheline! = !,labelasB’b. RotatepointB180oabouttheorigin,labelasB’’

c. TranslatepointBthepointup3andright7units,

labelasB’’’

-5

-5

5

5

A

y = x-5

-5

5

5

B

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6.1

GO Topic:Graphinglinearequations.

Grapheachfunctiononthecoordinategridprovided.Extendthelineasfarasthegridwillallow.

9. !(!) = 2! − 3 10. !(!) = −2! − 311. Whatsimilaritiesanddifferences

aretherebetweenthefunctionsf(x)

andg(x)?

12. ℎ(!) = !! ! + 1 13. !(!) = − !

! ! + 1

14. Whatsimilaritiesanddifferences

aretherebetweentheequationsh(x)

andk(x)?

15. !(!) = ! + 1 16. !(!) = ! − 317. Whatsimilaritiesanddifferences

aretherebetweentheequationsa(x)

andb(x)?

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

-5

-5

5

5

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TRANSFORMATIONS AND SYMMETRY – 6.3




6. 3 Leap Frog

A Solidify Understanding Task

JoshisanimatingasceneinwhichatroupeoffrogsisauditioningfortheAnimalChannel

realityshow,"TheBayou'sGotTalent".Inthisscenethefrogsaredemonstratingtheir"leapfrog"

acrobaticsact.Joshhascompletedafewkeyimagesinthissegment,andnowneedstodescribethe

transformationsthatconnectvariousimagesinthescene.

Foreachpre-image/imagecombinationlistedbelow,describethetransformationthat

movesthepre-imagetothefinalimage.

• Ifyoudecidethetransformationisarotation,youwillneedtogivethecenterofrotation,thedirectionoftherotation(clockwiseorcounterclockwise),andthemeasureoftheangleofrotation.

• Ifyoudecidethetransformationisareflection,youwillneedtogivetheequationofthelineofreflection.

• Ifyoudecidethetransformationisatranslationyouwillneedtodescribethe"rise"and"run"betweenpre-imagepointsandtheircorrespondingimagepoints.

• Ifyoudecideittakesacombinationoftransformationstogetfromthepre-imagetothefinalimage,describeeachtransformationintheordertheywouldbecompleted.

Pre-image FinalImage Description

image1 image2

image2 image3

image3 image4

image1 image5

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images this page:

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CC0 http://openclipart.org/detail/170806/hatalar205

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6.3

READY

Topic:RotationsandReflectionsoffigures.

Ineachproblemtherewillbeapre-imageandseveralimagesbasedonthegivepre-image.Determinewhichoftheimagesarerotationsofthegivenpre-imageandwhichofthemarereflectionsofthepre-image.Ifanimageappearstobecreatedastheresultofarotationandareflectionthenstateboth.(Compareallimagestothepre-image.)

1.

2.


ImageA ImageB ImageC ImageD

ImageA

ImageB

ImageC

ImageD

Pre-Image

Pre-Image

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6.3

SET Topic:Reflectingandrotatingpoints.Oneachofthecoordinategridsthereisalabeledpointandline.Usethelineasalineofreflectiontoreflectthegivenpointandcreateitsreflectedimageoverthelineofreflection.(Hint:pointsreflectalongpathsperpendiculartothelineofreflection.Useperpendicularslope!)3. 4.

5. 6.

Foreachpairofpoint,PandP’drawinthelineofreflectionthatwouldneedtobeusedtoreflectPontoP’.Thenfindtheequationofthelineofreflection.

7. 8.

m-5

-5

5

5A

l

-5

-5

5

5

C

ReflectpointAoverlinemandlabeltheimageA’ ReflectpointBoverlinekandlabeltheimageB’

ReflectpointCoverlinelandlabeltheimageC’ ReflectpointDoverlinemandlabeltheimageD’

k

-5

-5

5

5

B

m

-5

-5

5

5 D

-5

-5

5

5

P'

P-5

-5

5

5

P'

P

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6.3

Foreachpairofpoint,AandA’drawinthelineofreflectionthatwouldneedtobeusedtoreflectAontoA’.Thenfindtheequationofthelineofreflection.9. 10.

GO Topic:Slopesofparallelandperpendicularlinesandfindingslopeanddistancebetweentwopoints.

Foreachlinearequationwritetheslopeofalineparalleltothegivenline.11. ! = −3! + 5 12. ! = 7! – 3 13. 3! − 2! = 8

Foreachlinearequationwritetheslopeofalineperpendiculartothegivenline.14. ! = − !

! ! + 5 15. ! = !! ! − 4 16. 3! + 5! = −15

Findtheslopebetweeneachpairofpoints.Then,usingthePythagoreanTheorem,findthedistancebetweeneachpairofpoints.Youmayusethegraphtohelpyouasneeded.

17. (-2,-3)(1,1)a. Slope: b. Distance:

18. (-7,5)(-2,-7)a. Slope: b. Distance:

A'

A

A'

A

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6. 4 Leap Year

A Practice Understanding Task

CarlosandClaritaarediscussingtheirlatestbusinessventurewiththeirfriendJuanita.

Theyhavecreatedadailyplannerthatisbotheducationalandentertaining.Theplannerconsistsof

apadof365pagesboundtogether,onepageforeachdayoftheyear.Theplannerisentertaining

sinceimagesalongthebottomofthepagesformaflip-bookanimationwhenthumbedthrough

rapidly.Theplanneriseducationalsinceeachpagecontainssomeinterestingfacts.Eachmonth

hasadifferenttheme,andthefactsforthemonthhavebeenwrittentofitthetheme.Forexample,

thethemeforJanuaryisastronomy,thethemeforFebruaryismathematics,andthethemefor

Marchisancientcivilizations.CarlosandClaritahavelearnedalotfromresearchingthefactsthey

haveincluded,andtheyhaveenjoyedcreatingtheflip-bookanimation.

Thetwinsareexcitedto

sharetheprototypeof

theirplannerwithJuanita

beforesendingitto

printing.Juanita,

however,hasamajor

concern."Nextyearis

leapyear,"sheexplains,

"youneed366pages."

SonowCarlosandClarita

havethedilemmaof

needingtocreateanextra

pagetoinsertbetween

February28andMarch1.

Herearetheplanner

pagestheyhavealready

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Part1

SincethethemeforthefactsforFebruaryismathematics,Claritasuggeststhattheywrite

formaldefinitionsofthethreerigid-motiontransformationstheyhavebeenusingtocreatethe

imagesfortheflip-bookanimation.

Howwouldyoucompleteeachofthefollowingdefinitions?

1. Atranslationofasetofpointsinaplane...

2. Arotationofasetofpointsinaplane...

3. Areflectionofasetofpointsinaplane...

4. Translations,rotationsandreflectionsarerigidmotiontransformationsbecause...

CarlosandClaritausedthesewordsandphrasesintheirdefinitions:perpendicular

bisector,centerofrotation,equidistant,angleofrotation,concentriccircles,parallel,image,pre-

image,preservesdistanceandanglemeasureswithintheshape.Reviseyourdefinitionssothat

theyalsousethesewordsorphrases.

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Part2

InadditiontowritingnewfactsforFebruary29,thetwinsalsoneedtoaddanotherimage

inthemiddleoftheirflip-bookanimation.TheanimationsequenceisofDorothy'shousefromthe

WizardofOzasitisbeingcarriedovertherainbowbyatornado.ThehouseintheFebruary28

drawinghasbeenrotatedtocreatethehouseintheMarch1drawing.Carlosbelievesthathecan

getfromtheFebruary28drawingtotheMarch1drawingbyreflectingtheFebruary28drawing,

andthenreflectingitagain.

VerifythattheimageCarlosinsertedbetweenthetwoimagesthatappearedonFebruary28

andMarch1worksasheintended.Forexample,

• WhatconvincesyouthattheFebruary29imageisareflectionoftheFebruary28imageaboutthegivenlineofreflection?

• WhatconvincesyouthattheMarch1imageisareflectionoftheFebruary29imageaboutthegivenlineofreflection?

• WhatconvincesyouthatthetworeflectionstogethercompletearotationbetweentheFebruary28andMarch1images?

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Images this page:

http://openclipart.org/detail/168722/simple-farm-pack-by-viscious-speed

www.clker.com/clipart.tornado-gray

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6.4

READY

Topic:Definingpolygonsandtheirattributes

Foreachofthegeometricwordsbelowwriteadefinitionoftheobjectthataddressestheessentialelements.

1. Quadrilateral:

2. Parallelogram:

3. Rectangle:

4. Square:

5. Rhombus:

6. Trapezoid:

SET Topic:Reflectionsandrotations,composingreflectionstocreatearotation.

7.


UsethecenterofrotationpointCandrotatepoint

Pclockwisearoundit900.LabeltheimageP’.

WithpointCasacenterofrotationalsorotate

pointP1800.LabelthisimageP’’.C

P

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6.4

8.

9.

10.

11.

UsethecenterofrotationpointCandrotatepointPclockwisearoundit900.LabeltheimageP’.WithpointCasacenterofrotationalsorotatepointP1800.LabelthisimageP’’.

a. Whatistheequationforthelineforreflectionthat

reflectspointPontoP’?b. Whatistheequationforthelineofreflectionsthat

reflectspointP’ontoP’’?c. CouldP’’alsobeconsideredarotationofpointP?Ifsowhatisthecenterofrotationandhowmanydegreeswas

pointProtated?



reflectspointP’ontoP’’?c. CouldP’’alsobeconsideredarotationofpointP?Ifsowhatisthecenterofrotationandhowmanydegrees

waspointProtated?



reflectspointP’ontoP’’?c. CouldP’’alsobeconsideredarotationofpointP?Ifsowhatisthecenterofrotationandhowmanydegreeswas

pointProtated?

P

C

P''

P'P

P''

P'

P

P''

P'

P

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6.4

GO Topic:Rotationsabouttheorigin.

Plotthegivencoordinateandthenperformtheindicatedrotationinaclockwisedirectionaroundtheorigin,thepoint(0,0),andplottheimagecreated.Statethecoordinatesoftheimage.

12. PointA(4,2)rotate1800 13. PointB(-5,-3)rotate900clockwiseCoordinatesforPointA’(___,___) CoordinatesforPointB’(___,___)

14. PointC(-7,3)rotate1800 15. PointD(1,-6)rotate900clockwiseCoordinatesforPointC’(___,___) CoordinatesforPointD’(___,___)

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6. 5 Symmetries of Quadrilaterals

A Develop Understanding Task

Alinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbe

carriedontoitselfbyarotationissaidtohaverotationalsymmetry.

Everyfour-sidedpolygonisaquadrilateral.Somequadrilateralshaveadditionalproperties

andaregivenspecialnameslikesquares,parallelogramsandrhombuses.Adiagonalofa

quadrilateralisformedwhenoppositeverticesareconnectedbyalinesegment.Somequadrilaterals

aresymmetricabouttheirdiagonals.Somearesymmetricaboutotherlines.Inthistaskyouwilluse

rigid-motiontransformationstoexplorelinesymmetryandrotationalsymmetryinvarioustypesof

quadrilaterals.

Foreachofthefollowingquadrilateralsyouaregoingtotrytoanswerthequestion,“Isit

possibletoreflectorrotatethisquadrilateralontoitself?”Asyouexperimentwitheachquadrilateral,

recordyourfindingsinthefollowingchart.Beasspecificaspossiblewithyourdescriptions.

Definingfeaturesofthequadrilateral

Linesofsymmetrythatreflectthequadrilateral

ontoitself

Centerandanglesofrotationthatcarrythequadrilateral

ontoitself

Arectangleisaquadrilateralthatcontainsfourrightangles.

Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.

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Atrapezoidisaquadrilateralwithonepairofoppositesidesparallel.Isitpossibletoreflectorrotateatrapezoidontoitself?Drawatrapezoidbasedonthisdefinition.Thenseeifyoucanfind:

• anylinesofsymmetry,or• anycentersofrotationalsymmetry,

thatwillcarrythetrapezoidyoudrewontoitself.

Ifyouwereunabletofindalineofsymmetryoracenterofrotationalsymmetryforyourtrapezoid,seeifyoucansketchadifferenttrapezoidthatmightpossesssometypeofsymmetry.

Arhombusisaquadrilateralinwhichallsidesarecongruent.

Asquareisbotharectangleandarhombus.

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6.5

READY

Topic:Polygons,definitionandnames

1. Whatisapolygon?Describeinyourownwordswhatapolygonis.

2. Fillinthenamesofeachpolygonbasedonthenumberofsidesthepolygonhas.

NumberofSides NameofPolygon

3

4

5

6

7

8

9

10

SET Topic:Kites,Linesofsymmetryanddiagonals.

3. Onequadrilateralwithspecialattributesisakite.Findthegeometricdefinitionofakiteandwriteitbelowalongwithasketch.(Youcandothisfairlyquicklybydoingasearchonline.)

4. Drawakiteanddrawallofthelinesofreflectivesymmetryandallofthediagonals.

LinesofReflectiveSymmetry Diagonals


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6.5

5. Listalloftherotationalsymmetryforakite.

6. Arelinesofsymmetryalsodiagonalsinapolygon?Explain.

6. Arealldiagonalsalsolinesofsymmetryinapolygon?Explain.

7. Whichquadrilateralshavediagonalsthatarenotlinesofsymmetry?Namesomeanddrawthem.

8. Doparallelogramshavediagonalsthatarelinesofsymmetry?Ifso,drawandexplain.Ifnotdrawand

explain.

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6.5

GO Topic:Equationsforparallelandperpendicularlines.

FindtheequationofalinePARALLELtothegiveninfoandthroughtheindicated

y-intercept.

FindtheequationofalinePERPENDICULARtothe

givenlineandthroughtheindicatedy-intercept.

9. Equationofaline:! = 4! + 1.

a. Parallellinethroughpoint(0,-7):

b. Perpendiculartothelinelinethroughpoint(0,-7):

10. Tableofaline:

x y3 -84 -105 -126 -14

a. Parallellinethroughpoint(0,8):

b. Perpendiculartothelinethroughpoint(0,8):

11. Graphofaline: a. Parallellinethroughpoint(0,-9):

b. Perpendiculartothelinethroughpoint(0,-9):

-5

-5

5

5

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6.6 Symmetries of Regular

Polygons

A Solidify Understanding Task

Alinethatreflectsafigureontoitselfiscalledalineofsymmetry.Afigurethatcanbecarriedonto

itselfbyarotationissaidtohaverotationalsymmetry.Adiagonalofapolygonisanyline

segmentthatconnectsnon-consecutiveverticesofthepolygon.

Foreachofthefollowingregularpolygons,describetherotationsandreflectionsthatcarryitonto

itself:(beasspecificaspossibleinyourdescriptions,suchasspecifyingtheangleofrotation)

2. Asquare

3. AregularpentagonC

C B

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4. Aregularhexagon

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5. Aregularoctagon 6. Aregularnonagon

Whatpatternsdoyounoticeintermsofthenumberandcharacteristicsofthelinesofsymmetryina

regularpolygon?

Whatpatternsdoyounoticeintermsoftheanglesofrotationwhendescribingtherotational

symmetryinaregularpolygon?

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6.6

READY

Topic:Rotationalsymmetry,connectedtofractionsofaturnanddegrees.

1. Whatfractionofaturndoesthewagonwheel

belowneedtoturninordertoappearthevery

sameasitdoesrightnow?Howmanydegreesof

rotationwouldthatbe?

2. Whatfractionofaturndoesthepropeller

belowneedtoturninordertoappearthevery

sameasitdoesrightnow?Howmanydegreesof


3. WhatfractionofaturndoesthemodelofaFerriswheelbelowneedtoturninordertoappearthe

verysameasitdoesrightnow?Howmanydegreesof



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6.6

SET Topic:Findinganglesofrotationalsymmetryforregularpolygons,linesofsymmetryanddiagonals

4. Drawthelinesofsymmetryforeachregularpolygon,fillinthetableincludinganexpressionforthe

numberoflinesofsymmetryinan-sidedpolygon.

5. Drawallofthediagonalsineachregularpolygon.Fillinthetableandfindapattern,isitlinear,

exponentialorneither?Howdoyouknow?Attempttofindanexpressionforthenumberofdiagonalsin

an-sidedpolygon.

Numberof

Sides

Numberoflines

ofsymmetry

3

4

5

6

7

8

n

Numberof

Sides

Numberof

diagonals

3

4

5

6

7

8

n

Page 30






6.6

6. Findtheangle(s)ofrotationthatwillcarrythe12sidedpolygonbelowontoitself.

7. Whataretheanglesofrotationfora20-gon?Howmanylinesofsymmetry(linesofreflection)willit

have?

8. Whataretheanglesofrotationfora15-gon?Howmanylineofsymmetry(linesofreflection)willit

have?

9. Howmanysidesdoesaregularpolygonhavethathasanangleofrotationequalto180?Explain.

10. Howmanysidesdoesaregularpolygonhavethathasanangleofrotationequalto200?Howmany

linesofsymmetrywillithave?

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6.6

GO Topic:Reflectingandrotatingpointsonthecoordinateplane.

(Thecoordinategrid,compass,rulerandothertoolsmaybehelpfulindoingthiswork.)

9. ReflectpointAoverthelineofreflectionandlabeltheimageA’.

10. ReflectpointAoverthelineofreflectionandlabeltheimageA’.

11. ReflecttriangleABCoverthelineofreflectionandlabeltheimageA’B’C’.

12. ReflectparallelogramABCDoverthelineofreflectionandlabeltheimageA’B’C’D’.

13. GiventriangleXYZanditsimageX’Y’Z’drawthelineofreflectionthatwasused.

14GivenparallelogramQRSTanditsimageQ’R’S’T’drawthelineofreflectionthatwasused.

Z'

Y'

X'

ZY

X

R'

Q'T'T

Q

S S'R

line of reflection

A

line of reflection

A

line of reflection

AC

B

line of reflectionA

CB

D

Page 32






6.7 Quadrilaterals—Beyond

Definition

A Practice Understanding Task

Wehavefoundthatmanydifferentquadrilateralspossesslinesofsymmetryand/orrotational

symmetry.Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedin

termsoftheirsymmetries.

Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheirsymmetriesand

highlightedinthestructureoftheabovechart?

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Basedonthesymmetrieswehaveobservedinvarioustypesofquadrilaterals,wecanmakeclaims

aboutotherfeaturesandpropertiesthatthequadrilateralsmaypossess.

1. Arectangleisaquadrilateralthatcontainsfourrightangles.

Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutrectanglesbesidesthe

definingpropertythat“allfouranglesarerightangles?”Makealistofadditionalpropertiesof

rectanglesthatseemtobetruebasedonthetransformation(s)oftherectangleontoitself.Youwill

wanttoconsiderpropertiesofthesides,theangles,andthediagonals.Thenjustifywhythe

propertieswouldbetrueusingthetransformationalsymmetry.

2. Aparallelogramisaquadrilateralinwhichoppositesidesareparallel.

Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutparallelogramsbesides

thedefiningpropertythat“oppositesidesofaparallelogramareparallel?”Makealistofadditional

propertiesofparallelogramsthatseemtobetruebasedonthetransformation(s)ofthe

parallelogramontoitself.Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.

Thenjustifywhythepropertieswouldbetrueusingthetransformationalsymmetry.

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3. Arhombusisaquadrilateralinwhichallfoursidesarecongruent.

Basedonwhatyouknowabouttransformations,whatelsecanwesayaboutarhombusbesidesthe

definingpropertythat“allsidesarecongruent?”Makealistofadditionalpropertiesofrhombuses

thatseemtobetruebasedonthetransformation(s)oftherhombusontoitself.Youwillwantto

considerpropertiesofthesides,anglesandthediagonals.Thenjustifywhythepropertieswouldbe

trueusingthetransformationalsymmetry.

4. Asquareisbotharectangleandarhombus.

Basedonwhatyouknowabouttransformations,whatcanwesayaboutasquare?Makealistof

propertiesofsquaresthatseemtobetruebasedonthetransformation(s)ofthesquaresontoitself.

Youwillwanttoconsiderpropertiesofthesides,anglesandthediagonals.Thenjustifywhythe

propertieswouldbetrueusingthetransformationalsymmetry.

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Inthefollowingchart,writethenamesofthequadrilateralsthatarebeingdescribedintermsof

theirfeaturesandproperties,andthenrecordanyadditionalfeaturesorpropertiesofthattypeof

quadrilateralyoumayhaveobserved.Bepreparedtosharereasonsforyourobservations.

Whatdoyounoticeabouttherelationshipsbetweenquadrilateralsbasedontheircharacteristics

andthestructureoftheabovechart?

Howarethechartsatthebeginningandendofthistaskrelated?Whatdotheysuggest?

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6.7

READY

Topic:Definingcongruenceandsimilarity.

1. Whatdoyouknowabouttwofiguresiftheyarecongruent?

2. Whatdoyouneedtoknowabouttwofigurestobeconvincedthatthetwofiguresarecongruent?

3. Whatdoyouknowabouttwofiguresiftheyaresimilar?

4. Whatdoyouneedtoknowabouttwofigurestobeconvincedthatthetwofiguresaresimilar?

SET Topic:Classifyingquadrilateralsbasedontheirproperties.

Usingtheinformationgivendeterminethemostaccurateclassificationofthequadrilateral.

5. Has1800rotationalsymmetry. 6. Has900rotationalsymmetry.

7. Hastwolinesofsymmetrythatarediagonals. 8. Hastwolinesofsymmetrythatarenotdiagonals.

9. Hascongruentdiagonals. 10. Hasdiagonalsthatbisecteachother.

11. Hasdiagonalsthatareperpendicular. 12. Hascongruentangles.


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6.7

GO Topic:Slopeanddistance.

Findtheslopebetweeneachpairofpoints.Then,usingthePythagoreanTheorem,findthedistancebetweeneachpairofpoints.Distancesshouldbeprovidedinthemostexactform.

13. (-3,-2),(0,0)

a. Slope: b. Distance:

14. (7,-1),(11,7)


15. (-10,13),(-5,1)


16. (-6,-3),(3,1)


17. (5,22),(17,28)


18. (1,-7),(6,5)


S

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