similarity & right triangle trigonometry...secondary math ii // module 6 similarity & right...

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

© 2017 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 8: MVP MODULE 6

Similarity & Right Triangle Trigonometry

SECONDARY

MATH TWO

An Integrated Approach

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SECONDARY MATH II // MODULE 6

SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY

Mathematics Vision Project

Licensed under the Creative Commons Attribution CC BY 4.0

mathematicsvisionproject.org


MODULE 6 - TABLE OF CONTENTS

Similarity & Right Triangle Trigonometry

6.1 Photocopy Faux Pas – A Develop Understanding Task

Describing the essential features of a dilation (NC.M3.G-SRT.1, NC.M3.F-IF.2)

READY, SET, GO Homework: Similarity & Right Triangle Trigonometry 6.1

6.2 Triangle Dilations – A Solidify Understanding Task

Examining proportionality relationships in triangles that are known to be similar to each other based

on dilations (NC.M3.G-SRT.2a, NC.M3.G-SRT.4, NC.M3.G-CO.210, NC.M3.F-IF.2, )


6.3 Similar Triangles and Other Figures – A Solidify Understanding Task

Comparing definitions of similarity based on dilations and relationships between corresponding sides

and angles (NC.M3.G-SRT.2a,b, NC.M3.G-SRT.3)


6.4 Cut by a Transversal – A Solidify Understanding Task

Examining proportionality relationships of segments when two transversals intersect sets of parallel

lines (NC.M3.G-SRT.4)


6.5 Measured Reasoning – A Practice Understanding Task

Applying theorems about lines, angles and proportional relationships when parallel lines are crossed

by multiple transversals (NC.M3.G-CO.9, NC.M3.G-CO.10, NC.M3.G-SRT.4)


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SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY





6.6 Yard Work in Segments – A Solidify Understanding Task

Applying understanding of similar and congruent triangles to find the midpoint or any point on a

line segment that partitions the segment into a given ratio (NC.M3.G-GPE.6)


6.7 Pythagoras by Proportions – A Practice Understanding Task

Using similar triangles to prove the Pythagorean theorem and theorems about geometric means in

right triangles (NC.M3.G-SRT.4, NC.M3.G-SRT.5)


6.8 Are Relationships Predictable? – A Develop Understanding Task

Developing an understanding of right triangle trigonometric relationships based on similar

triangles (NC.M3.G-SRT.6, NC.M3.G-SRT.8, NC.M3.A-SSE.1a, NC.M3.A-CED.1)


5.5 Special Rights – A Solidify Understanding Task

Examining the relationship of sides in special right triangles (NC.M3.G-SRT.12)

READY, SET, GO Homework: Modeling with Geometry 5.5

6.10 Finding the Value of a Relationship – A Solidify Understanding Task

Solving for unknowns in right triangles using trigonometric ratios

(NC.M3.G-SRT.8, NC.M3.A-CED.1, NC.M3.A-SSE.1a)


6.11 Solving Right Triangles Using Trigonometric Relationships –

A Practice Understanding Task

Setting up and solving right triangles to model real world contexts

(NC.M3.G-SRT.8, NC.M3.G-SRT.12, NC.M3.A-CED.1, NC.M3.A.SSE.1a)


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SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.1

Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

6. 1 Photocopy Faux Pas

A Develop Understanding Task

Burnellhasanewjobatacopycenterhelpingpeopleusethe

photocopymachines.Burnellthinksheknowseverythingaboutmakingphotocopies,andsohe

didn’tcompletehisassignmenttoreadthetrainingmanual.

Mr.andMrs.DonahuearemakingascrapbookforMr.Donahue’sgrandfather’s75th

birthdayparty,andtheywanttoenlargeasketchoftheirgrandfatherwhichwasdrawnwhenhe

wasinWorldWarII.Theyhavepurchasedsomeveryexpensivescrapbookpaper,andtheywould

likethisimagetobecenteredonthepage.Becausetheyareunfamiliarwiththeprocessof

enlarginganimage,theyhavecometoBurnellforhelp.

“Wewouldliketomakeacopyofthisimagethatistwiceasbig,andcenteredinthemiddle

ofthisveryexpensivescrapbookpaper,”Mrs.Donahuesays.“Canyouhelpuswiththat?”

“Certainly,”saysBurnell.“Gladtobeofservice.”

Burnelltapedtheoriginalimageinthemiddleofawhitepieceofpaper,placeditonthe

glassofthephotocopymachine,insertedtheexpensivescrapbookpaperintothepapertray,and

settheenlargementfeatureat200%.

Inamoment,thisimagewasproduced.

“You’veruinedourexpensivepaper,”criedMrs.Donahue.

“Muchoftheimageisoffthepaperinsteadofbeingcentered.”

“Andthisimageismorethantwiceasbig,”Mr.Donahuecomplained.“Onefourthof

grandpa’spictureistakingupasmuchspaceastheoriginal.”

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Inthediagrambelow,boththeoriginalimage—whichBurnelltapedinthemiddleofasheetof

paper—andthecopyoftheimagehavebeenreproducedinthesamefigure.

1. Explainhowthephotocopymachineproducedthepartialcopyoftheoriginalimage.

2. Usinga“rubberbandstretcher”finishtherestoftheenlargedsketch.

3. WhereshouldBurnellhaveplacedtheoriginalimageifhewantedthefinalimagetobecenteredonthepaper?

4. Mr.Donahuecomplainedthatthecopywasfourtimesbiggerthantheoriginal.Whatdoyouthink?DidBurnelldoubletheimageorquadrupleit?Whatevidencewouldyouusetosupportyourclaim?

5. Transformingafigurebyshrinkingorenlargingitinthiswayisacalledadilation.Basedonyourthinkingabouthowthephotocopywasproduced,listallofthethingsyouneedtopayattentiontowhendilatinganimage.

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6.1

Needhelp?Visitwww.rsgsupport.org

READY Topic:Scalefactorsforsimilarshapes

Givethefactorbywhicheachpre-imagewasmultipliedtocreatetheimage.Usethescalefactortofillinanymissinglengths.

1. 2.

3. 4.

5. 6.

READY, SET, GO! Name Period Date

Pre-image

Image

Pre-image

Image

Pre-image

Pre-image Image

Image

Image: Large Triangle

Pre-Image:

Small Right Triangle Pre-Image

Image

10

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6.1


SETTopic:DilationsinrealworldcontextsForeachreal-worldcontextorcircumstancedeterminethecenterofthedilationandthetoolbeingusedtodothedilation.

7. 8.Franwalksbackwardtoadistancethatwillallowherfamilytoallshowupinthephotosheisabouttotake.

Thetheatretechnicianplayswiththezoominandoutbuttonsinefforttofilltheentiremoviescreenwiththeimage.

9. 10.Melanieestimatestheheightofthewaterfallbyholdingoutherthumbandusingittoseehowmanythumbstalltothetopofthewaterfallfromwheresheisstanding.Shethenusesherthumbtoseethatapersonatthebaseofthewaterfallishalfathumbtall.

Adigitalanimatorcreatesartisticworksonhercomputer.Sheiscurrentlydoingananimationthathasseveraltelephonepolesalongastreetthatgoesoffintothedistance.

11. 12.Ms.Sunshineishavingherclassdoaprojectwitharubber-bandtracingdevicethatusesthreerubberbands.

Acopymachineissetat300%formakingaphotocopy.

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6.1


GO Topic:Ratesofchangerelatedtolinear,exponentialandquadraticfunctions

Determinewhetherthegivenrepresentationisrepresentativeofalinear,exponentialorquadraticfunction,classifyassuchandjustifyyourreasoning.13. 14.

Typeoffunction:Justification:

X Y2 73 124 195 28


15. ! = 3!! + 3! 16. ! = 7! − 10Typeoffunction:Justification:


17. 18.

Typeoffunction:

Justification:

Typeoffunction:

Justification:

X Y2 -57 514 1925 41

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6. 2 Triangle Dilations

A Solidify Understanding Task

1. GivenΔABC,usepointMasthecenterofadilationtolocatetheverticesofatrianglethathas

sidelengthsthatarethreetimeslongerthanthesidesofΔABC.

2. NowusepointNasthecenterofadilationtolocatetheverticesofatrianglethathasside

lengthsthatareone-halfthelengthofthesidesofΔABC.

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3. Labeltheverticesinthetwotrianglesyoucreatedinthediagramabove.Basedonthisdiagram,

writeseveralproportionalitystatementsyoubelievearetrue.Firstwriteyourproportionality

statementsusingthenamesofthesidesofthetrianglesinyourratios.Thenverifythatthe

proportionsaretruebyreplacingthesidenameswiththeirmeasurements,measuredtothe

nearestmillimeter.

Mylistofproportions:(trytofindatleast10proportionalitystatementsyoubelieveare

true)

4. Basedonyourworkabove,underwhatconditionsarethecorrespondinglinesegmentsinan

imageanditspre-imageparallelafteradilation?Thatis,whichwordbestcompletesthis

statement?

Afteradilation,correspondinglinesegmentsinanimageanditspre-imageare[never,

sometimes,always]parallel.

5. Givereasonsforyouranswer.Ifyouchoose“sometimes”,beveryclearinyourexplanation

abouthowyoucantotellwhenthecorrespondinglinesegmentsbeforeandafterthedilation

areparallelandwhentheyarenot.

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GivenΔABC,usepointAasthecenterofadilationtolocatetheverticesofatrianglethathasside

lengthsthataretwiceaslongasthesidesofΔABC.

6. Explainhowthediagramyoucreatedabovecanbeusedtoprovethefollowingtheorem:

Thesegmentjoiningmidpointsoftwosidesofatriangleisparalleltothethirdsideandhalfthe

length.

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6.2


READY Topic:Basicanglerelationships

Matchthediagramsbelowwiththebestnameorphrasethatdescribestheangles.______1. _______2.

______3. ______4.

_____5. ______6.

a. AlternateInteriorAngles b. VerticalAngles c. ComplementaryAngles

d. TriangleSumTheorem e. LinearPair f. SameSideInteriorAngles


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6.2


SETTopic:Performingmathematicaldilationsandfindingthecenterofdilations.

Usethegivenpre-imageandpointCasthecenterofdilationtoperformthedilationthatisindicatedbelow.

7. Createanimagewithsidelengthstwicethesizeofthegiventriangle.

8. Createanimagewithsidelengthshalfthesizeofthegiventriangle.

9. Createanimagewithsidelengthsthreetimesthesizeofthegivenparallelogram.

10. Createanimagewithsidelengthonefourththesizeofthegivenpentagon.

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6.2


Usethegivenpre-imageandimageineachdiagramtodefinethedilationthatoccurred.Includeasmanydetailsaspossiblesuchasthecenterofthedilationandtheratio.

11. 12.

13. 14.

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6.2


GO Topic:Classifyingmathematicaltransformations.

Basedonthegivenimageandpre-imagedeterminethetransformationthatoccurred.Further,provethatthetransformationoccurredbyshowingevidenceofsomekind.

(Forexample,ifthetransformationwasareflectionshowthelineofreflectionexistsandprovethatitistheperpendicularbisectorofallsegmentsthatconnectcorrespondingpointsfromtheimageandpre-image.Dolikewiseforrotations,translationsanddilations.)

15. 16.

17. 18.

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6. 3 Similar Triangles and OtherFigures


Twofiguresaresaidtobecongruentifthesecondcanbeobtainedfromthefirstbyasequenceof

rotations,reflections,andtranslations.InMathematicsIwefoundthatweonlyneededthreepieces

ofinformationtoguaranteethattwotriangleswerecongruent:SSS,ASAorSAS.

WhataboutAAA?Aretwotrianglescongruentifallthreepairsofcorrespondinganglesare

congruent?Inthistaskwewillconsiderwhatistrueaboutsuchtriangles.

Part1

DefinitionofSimilarity:Twofiguresaresimilarifthesecondcanbeobtainedfromthefirst

byasequenceofrotations,reflections,translations,anddilations.

MasonandMiaaretestingoutconjecturesaboutsimilarpolygons.Hereisalistoftheir

conjectures.

Conjecture1:Allrectanglesaresimilar.

Conjecture2:Allequilateraltrianglesaresimilar.

Conjecture3:Allisoscelestrianglesaresimilar.

Conjecture4:Allrhombusesaresimilar.

Conjecture5:Allsquaresaresimilar.

1. Whichoftheseconjecturesdoyouthinkaretrue?Why?

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MasonisexplainingtoMiawhyhethinksconjecture1istrueusingthediagramgivenbelow.

“Allrectangleshavefourrightangles.IcantranslateandrotaterectangleABCDuntilvertexAcoincideswith

vertexQinrectangleQRST.Since∠A

and∠Qarebothrightangles,sideAB

willlieontopofsideQR,andsideAD

willlieontopofsideQT.Icanthen

dilaterectangleABCDwithpointAasthe

centerofdilation,untilpointsB,C,andD

coincidewithpointsR,S,andT.

2. DoesMason’sexplanationconvinceyouthatrectangleABCDissimilartorectangleQRSTbasedonthedefinitionofsimilaritygivenabove?Doeshisexplanationconvinceyouthatallrectanglesaresimilar?Whyorwhynot?

MiaisexplainingtoMasonwhyshethinksconjecture2istrueusingthediagramgivenbelow.

“Allequilateraltriangleshavethree60°angles.Ican

translateandrotateΔABCuntilvertexAcoincideswith

vertexQinΔQRS.Since∠Aand∠Qareboth60°

angles,sideABwilllieontopofsideQR,andsideAC

willlieontopofsideQS.IcanthendilateΔABCwith

pointAasthecenterofdilation,untilpointsBandC

coincidewithpointsRandS.”

3. DoesMia’sexplanationconvinceyouthatΔABCissimilartoΔQRSbasedonthedefinitionofsimilaritygivenabove?Doesherexplanationconvinceyouthatallequilateraltrianglesaresimilar?Whyorwhynot?

4. Foreachoftheotherthreeconjectures,writeanargumentlikeMason’sandMia’stoconvincesomeonethattheconjectureistrue,orexplainwhyyouthinkitisnotalwaystrue.

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a. Conjecture3:Allisoscelestrianglesaresimilar.

b. Conjecture4:Allrhombusesaresimilar.

c. Conjecture5:Allsquaresaresimilar.

Whilethedefinitionofsimilaritygivenatthebeginningofthetaskworksforallsimilarfigures,an

alternativedefinitionofsimilaritycanbegivenforpolygons:Twopolygonsaresimilarifall

correspondinganglesarecongruentandallcorrespondingpairsofsidesareproportional.

5. HowdoesthisdefinitionhelpyoufindtheerrorinMason’sthinkingaboutconjecture1?

6. HowdoesthisdefinitionhelpconfirmMia’sthinkingaboutconjecture2?

7. Howmightthisdefinitionhelpyouthinkabouttheotherthreeconjectures?

a. Conjecture3:Allisoscelestrianglesaresimilar.

b. Conjecture4:Allrhombusesaresimilar.

c. Conjecture5:Allsquaresaresimilar.

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Part2(AAA,SASandSSSSimilarity)

Fromourworkabovewithrectanglesitisobviousthatknowingthatallrectangleshavefourright

angles(anexampleofAAAAforquadrilaterals)isnotenoughtoclaimthatallrectanglesaresimilar.

Whatabouttriangles?Ingeneral,aretwotrianglessimilarifallthreepairsofcorrespondingangles

arecongruent?

8. Decideifyouthinkthefollowingconjectureistrue.

Conjecture:Twotrianglesaresimilariftheircorrespondinganglesarecongruent.

9. Explainwhyyouthinktheconjecture—twotrianglesaresimilariftheircorrespondinganglesarecongruent—istrue.Usethefollowingdiagramtosupportyourreasoning,Remembertostartbymarkingwhatyouaregiventobetrue(AAA)inthediagram.

Hint:BeginbytranslatingAtoD.

10. Miathinksthefollowingconjectureistrue.Shecallsit“AASimilarityforTriangles.”Whatdoyouthink?Isittrue?Why?

Conjecture:Twotrianglesaresimilariftheyhavetwopairofcorrespondingcongruentangles.

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11. Usingthediagramgiveninproblem9,howmightyoumodifyyourproofthatΔABC~ΔDEFifyouaregiventhefollowinginformationaboutthetwotriangles:

a. ∠A≅∠D,!" = ! ⋅ !", !" = ! ⋅ !" ;thatis, !"!" = !"!"

b. !" = ! ⋅ !", !" = ! ⋅ !" and !" = ! ⋅ !";thatis, !"!" =!"!" =

!"!"

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6.3


READY Topic:Solvingproportionsinmultipleways

Solveeachproportion.Showyourworkandcheckyoursolution.

1. 2. 3.34 =

!20

!7 =

1821

36 =

8!

4. 5. 6.9! =

610

34 =

! + 320

712 =

!24

7. 8. 9.!2 =

1320

3! + 2 =

65

32 = 12

!

SETTopic:ProvingShapesaresimilar

Provideanargumenttoproveeachconjecture,orprovideacounterexampletodisproveit.

10. Allrighttrianglesaresimilar 11. Allregularpolygonsaresimilartootherregular

polygonswiththesamenumberofsides.

12. Thepolygonsonthegridbelowaresimilar. 13. Thepolygonsonthegridbelowaresimilar.


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6.3


Asequenceoftransformationsoccurredtocreatethetwosimilarpolygons.Provideaspecificsetofstepsthatcanbeusedtocreatetheimagefromthepre-image.14. 15.

16. 17.

GO Topic:Ratiosinsimilarpolygons

Foreachpairofsimilarpolygonsgivethreeratiosthatwouldbeequivalent.18. 19.

20. 21.

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SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4 SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4



6. 4 Cut by a Transversal


Drawtwointersectingtransversalsonasheetoflinedpaper,asinthefollowingdiagram.LabelthepointofintersectionofthetransversalsA.Selectanytwoofthehorizontallinestoformthethirdsideoftwodifferenttriangles.

1. Whatconvincesyouthatthetwotrianglesformedbythetransversalsandthehorizontallinesaresimilar?

2. Labeltheverticesofthetriangles.Writesomeproportionalitystatementsaboutthesidesofthetrianglesandthenverifytheproportionalitystatementsbymeasuringthesidesofthetriangles.

3. Selectathirdhorizontallinesegmenttoformathirdtrianglethatissimilartotheothertwo.Writesomeadditionalproportionalitystatementsandverifythemwithmeasurements.

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SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4 SIMILARITY & RIGHT TRIANGLE TRIGONOMETRY – 6.4



Tristanhaswrittenthisproportionforquestion3,basedonhisdiagrambelow:!"!" =!"!"

TiathinksTristan’sproportioniswrong,becausesomeofthesegmentsinhisproportionarenotsidesofatriangle.

4. CheckoutTristan’sideausingmeasurementsofthesegmentsinhisdiagramattheleft.

5. Nowcheckoutthissameideausingproportionsofsegmentsfromyourowndiagram.Testatleasttwodifferentproportions,includingsegmentsthatdonothaveAasoneoftheirendpoints.

6. Basedonyourexamples,doyouthinkTristanorTiaiscorrect?

Tiastillisn’tconvinced,sinceTristanisbasinghisworkonasinglediagram.Shedecidestostart

withaproportionsheknowsistrue:!"!" =!"!".(Whyisthistrue?)

Tiarealizesthatshecanrewritethisproportionas!"!!"!" = !"!!"!" (Whyisthistrue?)

CanyouuseTia’sproportiontoprovealgebraicallythatTristaniscorrect?

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6.4


READY Topic:Pythagoreantheoremandproportionsinsimilartriangles.

Findthemissingsideineachrighttriangle

1. 2.

3. 4.

Createaproportionforeachsetofsimilartriangles.Thensolvetheproportion.

5. 6.

5

12?

3

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?4

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6.4


SETTopic:Proportionalityoftransversalsacrossparallellines

Forquestions7and8,writethreeequalratios.

7. Thelettersa,b,canddrepresentlengthsoflinesegments.

8.

9. Writeandsolveaproportionthatwillprovidethemissinglength.

10. Writeandsolveaproportionthatwillprovidethemissinglength.

Forquestions11–14findandlabeltheparallellines.(i.e.!" ∥ !")Thenwriteasimilaritystatementforthetrianglesthataresimilar.(i.e.∆ !"# ~ ∆ !"#)11. 12.

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6.4


13. 14.

GO Topic:Similarityinslopetriangles

Eachlinebelowhasseveraltrianglesthatcanbeusedtodeterminetheslope.Drawinthreeslope-definingtrianglesofdifferentsizesforeachlineandthencreatetheratioofrisetorunforeach.15. 16.

17. 18.

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6. 5 Measured Reasoning


Findthemeasuresofallmissingsidesandanglesbyusinggeometricreasoning,notrulersandprotractors.Ifyouthinkameasurementisimpossibletofind,identifywhatinformationyouaremissing.Linesp,q,r,andsareallparallel.

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1. Identifyatleastthreedifferentquadrilateralsinthediagram.Findthesumoftheinterioranglesforeachquadrilateral.Makeaconjectureaboutthesumoftheinterioranglesofaquadrilateral.

Conjecture:

2. Identifyatleastthreedifferentpentagonsinthediagram.(Hint:Thepentagonsdonotneedtobeconvex.)Findthesumoftheinterioranglesforeachpentagon.Makeaconjectureaboutthesumoftheinterioranglesofapentagon.

Conjecture:

3. Doyouseeapatterninthesumoftheanglesofapolygonasthenumberofsidesincreases?Howcanyoudescribethispatternsymbolically?

4. Howcanyouconvinceyourselfthatthispatternholdsforalln-gons?

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6.5


READY Topic:PythagoreanTheoremandratiosofsimilartrianglesFindthemissingsideineachrighttriangle.Trianglesarenotdrawntoscale.

7. Basedonratiosbetweensidelengths,whichoftherighttrianglesabovearemathematicallysimilartoeachother?Providethelettersofthetrianglesandtheratios.


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6.5


SET Topic:Usingparallellines,andanglerelationshipstofindmissingvalues.

Ineachofthediagramsusethegiveninformationprovidedtofindthemissinglengthsandanglemeasurements.

8. Linem∥nando∥p,findthevaluesofanglesx,yandz.Also,findthelengthsofa,bandc.

9. Lineq ∥r∥sandt∥uandp∥w ∥v,findthevaluesofanglesx,yandz.Also,findthelengthsofa,b,c,d,e,f.

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6.5


GO Topic:Solveequationsincludingthoseincludingproportions

Solveeachequationbelow.10. 11. 12.

3! − 5 = 2! + 7 57 =

!21

3! =

185! + 2

13. 14. 15.

12 ! − 7 =

34 ! − 8

17 + 3(! − 5) = 2(! + 3) ! + 56 = 3(! + 2)

9

16. 17. 18.

! + 2 + 3! − 8 = 90 512 =

!8

45 =

! + 215

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6. 6 Yard Work in Segments


Malik’sfamilyhaspurchasedanewhousewithanunfinishedyard.

Theydrewthefollowingmapofthebackyard:

Malikandhisfamilyareusingthemaptosetupgardensandpatiosfortheyard.Theyplantolay

outtheyardwithstakesandstringssotheyknowwheretoplantgrass,flowers,orvegetables.

Theywanttobeginwithavegetablegardenthatwillbeparalleltothefenceshownatthetopofthe

mapabove.

1. Theysetthefirststakeat(-9,6)andthestakeattheendofthegardenat(3,10).Theywanttomarkthemiddleofthegardenwithanotherstake.Whereshouldthestakethatisthemidpointofthesegmentbetweenthetwoendstakesbelocated?Usingadiagram,describeyourstrategyforfindingthispoint.

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Malikfiguredoutthemidpointbysaying,“Itmakessensetomethatthemidpointisgoingtobe

halfwayoverandhalfwayup,soIdrewarighttriangleandcutthehorizontalsideinhalfandthe

verticalsideinhalflikethis:”

Malikcontinued,“Thatputmerightat(-3,8).Theonlythingthatseemsfunnyaboutthattomeis

thatIknowthebaseofthebigtrianglewas12andtheheightofthetrianglewas4,soIthoughtthe

midpointmightbe(6,2).”

2. ExplaintoMalikwhythelogicthatmadehimthinkthemidpointwas(6,2)isalmostright,andhowtoextendhisthinkingtousethecoordinatesoftheendpointstogetthemidpointof(-3,6).

Malik’ssister,Sapana,lookedathisdrawingandsaid,“Hey,Idrewthesamepicture,butInoticed

thetwosmallertrianglesthatwereformedwerecongruent.SinceIdidn’tknowforsurewhatthe

midpointwas,Icalledit(x,y).ThenIusedthatpointtowriteanexpressionforthelengthofthe

sidesofthesmalltriangles.Forinstance,Ifiguredthatthebaseofthelowertrianglewasx–(-9).

3. LabelalloftheotherlegsofthetwosmallerrighttrianglesusingSapana’sstrategy.

(x,y)

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Sapanacontinued,“OnceIlabeledthetriangles,Iwroteequationsbymakingthebasesequalandtheheightsequal.”

4. DoesSapana’sstrategywork?Showwhyorwhynot.

5. Chooseastrategyanduseittofindthemidpointofthesegmentwithendpoints(-3,4)and(2,9).

6. Useeitherstrategytofindthemidpointofthesegmentbetween(x1,y1)and(x2,y2).

Thenextareainthegardentobemarkedisforaflowergarden.Malik’sparentshavetheideathat

partofthegardenshouldcontainabigrosebushandtherestofthegardenwillhavesmaller

flowerslikepetunias.Theywantthesectionwiththeotherflowerstobetwiceaslongasthe

sectionwiththerosebush.Thestakeontheendpointsofthis

gardenwillbeat(1,5)and(4,11).Malik’sdadsays,“We’llneed

astakethatmarkstheendoftherosegarden.”

7. HelpMalikandSapanafigureoutwherethestakewillbelocatediftherosebushwillbeclosertothestakeat(1,5)thanthestakeat(4,11).

(x1,y1)

(x2,y2)

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There’sonlyonemoresetofstakestoputin.Thistimetheendpointstakesareat(-8,5)and

(2,-10).Anotherstakeneedstobeplacedthatpartitionsthissegmentintotwopartssothatthe

ratioofthelengthsis2:3.

8. Wheremustthestakebelocatedifitistobeclosertothestakeat(-8,5)thantothestakeat(2,-10)?

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6.6


READY Topic:Averages,measuresofcenter,arithmeticmean

Foreachsetofnumbersfindthemean(average).Explainhowthemeanofthesetcomparestothevaluesintheset.1. 6,12,10,8 2. 2,7,12 3. -13,21

4. 3,-9,15 5. 43,52 6. 38,64,100

Findthevaluethatisexactlyhalfwaybetweenthetwogivenvalues.Explainhowyoufindthisvalue.7. 5,13 8. 26,42 9. 57,77

10. -34,-22 11. -45,3 12. -12,18

SETTopic:Midpointsofsegmentsandproportionalityofsidesinembeddedsimilartriangles

Findthecoordinatesofthemidpointofeachlinesegmentbelow.Ifmultiplelinesegmentsaregiventhengivethemidpointsofallsegments.13. 14.


Page 39






6.6


Useproportionalrelationshipstofindthedesiredvalues.

19. 20.

Ifalineisdrawnparallelto!"andthroughpointA.Atwhatcoordinatewilltheintersectionofthisparallellinebewith!"?

Ifalineisdrawnparallelto!"andthroughpointE.Atwhatcoordinatewilltheintersectionofthisparallellinebewith!"?

15. 16.

17. Alinesegmentbetween(2,3)and(10,15) 18.Alinesegmentbetween(-2,7)and(3,-8)

Page 40






6.6


21. 22.

Ifalineisdrawnparallelto!"andthroughpointF.Atwhatcoordinatewilltheintersectionofthisparallellinebewith!"?

Ifalineisdrawnparallelto!"andthroughpointG.Atwhatcoordinatewilltheintersectionofthisparallellinebewith!"?

23. Whenalineisdrawnparalleltoonesideofatrianglesothatitintersectstheothertwosidesofthetriangle,howdothemeasuresofthepartsofthetwointersectedsidescompare?Explain.

24. Problems19-22providedrighttriangles.Couldadeterminationofthecoordinatesbemadeiftheywerenotrighttriangles?Whyorwhynot?

GO Topic:Proportionalitywithparallellines.

Writeaproportionforeachofthediagramsbelowandsolveforthemissingvalue.25. 26.

Page 41




6. 7 Pythagoras by Proportions


TherearemanydifferentproofsofthePythagoreantheorem.Hereisonebasedonsimilar

triangles.

Step1:Cuta4×6indexcardalongoneofitsdiagonalstoformtwocongruentrighttriangles.

Step2:Ineachrighttriangle,drawanaltitudefromtherightanglevertextothehypotenuse.

Step3:Labeleachtriangleasshowninthefollowingdiagram.Flipeachtriangleoverandlabelthe

matchingsidesandangleswiththesamenamesonthebackasonthefront.

Step4:Cutoneoftherighttrianglesalongthealtitudetoformtwosmallerrighttriangles.

Step5:Arrangethethreetrianglesinawaythatconvincesyouthatallthreerighttrianglesare

similar.Youmayneedtoreflectand/orrotateoneormoretrianglestoformthisarrangement.

Step6:Writeproportionalitystatementstorepresentrelationshipsbetweenthelabeledsidesof

thetriangles.

Step7:Solveoneofyourproportionsforxandtheotherproportionfory.(Ifyouhavenotwritten

proportionsthatinvolvexandy,studyyoursetoftrianglesuntilyoucandoso.)

Step8:Workwiththeequationsyouwroteinstep7untilyoucanshowalgebraicallythat

!! + !! = !!.(Remember,x+y=c.)

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Part 1:




Useyoursetoftrianglestohelpyouprovethefollowingtwotheoremsalgebraically.Forthiswork,

youwillwanttolabelthelengthofthealtitudeoftheoriginalrighttriangleh.Theappropriatelegs

ofthesmallerrighttrianglesshouldalsobelabeledh.

RightTriangleAltitudeTheorem1:Ifanaltitudeisdrawntothehypotenuseofarighttriangle,

thelengthofthealtitudeisthegeometricmeanbetweenthelengthsofthetwosegmentsformedon

thehypotenuse.

RightTriangleAltitudeTheorem2:Ifanaltitudeisdrawntothehypotenuseofarighttriangle,

thelengthofeachlegoftherighttriangleisthegeometricmeanbetweenthelengthofthe

hypotenuseandthelengthofthesegmentonthehypotenuseadjacenttotheleg.

H: Useyoursetoftrianglestohelpyoufindthevaluesofxandyinthefollowingdiagram.

Page 44

Part 2H:






6.7


READY Topic:Determiningsimilarityandcongruenceintriangles

1. Determinewhichofthetrianglesbelowaresimilarandwhicharecongruent.Justifyyourconclusions.Giveyourreasoningforthetrianglesyoupicktobesimilarandcongruent.


Page 45






6.7


SETTopic:Similarityinrighttriangles

Usethegivenrighttriangleswithaltitudesdrawntothehypotenusetocorrectlycompletetheproportions.

2. 3.

!! =

!?

4H. !! =

!? 5H.

!! =

!?

6H.!! =

!? 7H.

!! =

!?

Findthemissingvalueforeachrighttrianglewithaltitude.Honors only should find the altitude.

9.

Page 46

!! =

!?

8.

HH






6.7


GO Topic:Usesimilarityandparallellinestosolveproblems.

Ineachproblemdeterminethedesiredvaluesusingthesimilartrianglesparallellinesandproportionalrelationships.Writeaproportionandsolve.

10. 11.

Analyzeeachtablebelowcloselyanddeterminethemissingvaluesbasedonthegiveninformationandvaluesinthetable.

12. AnArithmeticSequenceTerm 1 2 3 4Value 7 22

13. AGeometricSequenceTerm 1 2 3 4Value 7 56

14. AnArithmeticSequenceTerm 5 6 7 8Value 10 43

15. AGeometricSequenceTerm 7 8 9 10Value 3 24

Page 47




6. 8 Are Relationships Predictable?

A Develop Understanding Task

Inyournotebookdrawarighttrianglewithoneangleof60°.Measureeachsideofyourtriangleasaccuratelyasyoucanwithacentimeterruler.Usingthe600angleastheangleofreferencelistthemeasureforeachofthefollowing:

Lengthoftheadjacentside: Lengthoftheoppositeside:

Lengthofthehypotenuse:

Createthefollowingratiosusingyourmeasurements:

opposite sidehypotenuse

= adjacent sidehypotenuse

=

opposite sideadjacent side

=

1. Compareyourratioswithothersthathadatriangleofadifferentsize.Whatdoyounotice?Explainanyconnectionsyoufindtoothers’work?

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2. Intherighttrianglesbelowfindthemissingsidelengthandthencreatethedesiredratiosbasedontheangleofreference(angleAandangleD).

ListtheratiosforΔABC usingangleAastheangleofreference.

ListtheratiosforΔDEF usingangleDastheangleofreference.


=

adjacent sidehypotenuse

=


=


=

adjacent sidehypotenuse

=


=

3. Whatdoyounoticeabouttheratiosfromthetwogiventriangles?Howdotheseratioscomparetotheratiosfromthetriangleyoumadeonthepreviouspage?

4. Whatcanyouinferabouttheanglemeasuresof∆!"# and ∆!"#?

Page 50




5. Whydotherelationshipsyouhavenoticedoccur?

6. Whatcanyouconcludeabouttheratioofsidesinarighttrianglethathasa60°angle?Wouldyouthinkthatrighttriangleswithotheranglemeasureswouldhavesimilarrelationshipsamongtheirratios?

Page 51






6.8


READY Topic:Propertiesofanglesandsidesinrighttriangles

Foreachrighttrianglebelowfindthemissingsiden(PythagoreanTheoremcouldbehelpful)andthemissingangle,a(AngleSumTheoremforTrianglescouldbeuseful).1. 2. 3.

4. 5. 6.


Page 52






6.8


SETTopic:Creatingtrigonometricratiosforrighttriangles

Foreachrighttriangleandtheidentifiedangleofreferencecreatethedesiredtrigonometricratios.Ifanysidesofthetrianglearemissing,findthembeforedeterminingtheratio.

7. 8.

Cos(A)= Cos(B)= Cos(A)= Cos(B)=

Sin(A)= Sin(B)= Sin(A)= Sin(B)=

Tan(A)= Tan(B)= Tan(A)= Tan(B)=

9. 10.

Cos(A)= Cos(B)= Cos(A)= Cos(B)=

Sin(A)= Sin(B)= Sin(A)= Sin(B)=

Tan(A)= Tan(B)= Tan(A)= Tan(B)=

Page 53






6.8


GO Topic:Factoringquadratics

Providethefactoredformandthex-andy-intercepts

11. f(x)=x2+9x+20

factoredform:

x-intercepts:

y-intercept:

12. g(x)=x2+2x–15

factoredform:

x-intercepts:

y-intercept:

13. h(x)=x2–49

factoredform:

x-intercepts:

y-intercept:

14. r(x)=x2-13x+30

factoredform:

x-intercepts:

y-intercept:

15. f(x)=x2+20x+100

factoredform:

x-intercepts:

y-intercept:

16. g(x)=x2–8x–48

factoredform:

x-intercepts:

y-intercept:

17. h(x)=x2+16x+64

factoredform:

x-intercepts:

y-intercept:

18. k(x)=x2–36

factoredform:

x-intercepts:

y-intercept:

19. p(x)=x2–2x–24

factoredform:

x-intercepts:

y-intercept:

Page 54




6. 10 Finding the Value of a

Relationship


Part1:Pickaside

AndreaandBonitaarerestingundertheirfavoritetreebeforetakinganaturewalkupahill.Both

girlshavebeenstudyingtrigonometryinschool,andnowitseemsliketheyseerighttriangles

everywhere.Forexample,Andreanoticesthelengthoftheshadowofthetreetheyaresitting

underandwondersiftheycancalculatetheheightofthetreejustbymeasuringthelengthofits

shadow.

1. HowmightAndreaandBonitausethis

information,alongwiththeirknowledgeof

trigonometricratios,tocalculatetheheight

ofthetree?(AndreaandBonitaknowthey

canfindthevalueofanytrigonometric

ratiostheymightneedforanyacuteangle

usingacalculator.)

Bonitathinkstheyalsoneedtoknowthemeasureofanangle,soshe

checksanapponherphoneandfindsthattheangleofelevationofthesun

atthecurrentlocationandtimeofdayis50°.Inthemeantime,Andrea

haspacedoffthelengthofthetree’sshadowandfindsthatitis40feet

long.

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Page 55




Part2:What’syourangle?

Aftertheirrest,AndreaandBonitaaregoingforawalkstraightupthesideofthehill.Andrea

decidedtostretchbeforeheadingupthehillwhileBonitathoughtthiswouldbeagoodtimetogeta

headstart.OnceBonitawas100feetawayfromAndrea,shestoppedtotakeabreakandlookedat

herGPSdevicethattoldherthatshehadwalked100feetandhadalreadyincreasedherelevation

by40feet.Withabitoftimetowaste,Bonitawrotedownthetrigonometricratiosfor∠Aandfor

∠B.

2. Namethetrigonometricratiosfor∠Aandfor∠B.

WhenAndreacaughtup,shesaid“Whatabouttheunknownanglemeasures?WhenIwasatthe

bottomandlookeduptoseeyou,Iwasthinkingaboutthe“upward”anglemeasurefrommetoyou.

Basedonyourpicture,thiswouldbe∠A.”

Bonitawrotethetrigonometricratiosin! = !"!""andasked,“So,howdowefindangleA?”Together,thegirlstalkedabouthowthiswaslikethinkingbackwards:insteadofknowinganangle

andusingtheircalculatorstofindatrigonometricratioliketheydidwhileworkingontheheightof

thetreeproblem,theynowknowthetrigonometricratioandneedtofindanunknownanglevalue.

Bonitanoticesthe!"#!!(!)buttononhercalculatorandwondersifthismightworklikean“inversetrigonometricratio”button,undoingtheratiotoproducetheangle.Shedecidestotryit

out,andproducesthefollowingoutputonhercalculator:

sin! = !"!""

!"#!! 40100 = 23.578°

sin 23.578° = 0.4

3. HowmightthisoutputconvinceBonitathather

assumptionaboutthecalculatorwascorrect?

Page 56




4. Usethetrigonometricratioyoufoundforcos!tofindthevalueof∠B.

5. Findallunknownvaluesforthefollowingrighttriangle:

a) ∠! =______b) ∠! =______c) ∠!= 90°d) a=12m

e) b=8m

f) c=______

6. BonitaandAndreastartedtalkingaboutallofthewaystofindunknownvaluesinright

trianglesanddecidedtomakealist.Whatdoyouthinkshouldbeontheirlist?Bespecificand

preciseinyourdescription.Forexample,‘trigratios’isnotspecificenough.Youmayusethe

followingsentenceframetoassistwithwritingeachiteminyourlist:

Whengiven______________________,youcanfind_______________________by_________________.

Page 57




Part3:Angleofelevationandangleofdepression

Duringtheirhike,AndreamentionedthatshelookeduptoseeBonita.Inmathematics,whenyou

lookstraightahead,wesayyourlineofsightisahorizontalline.Fromthehorizontal,ifyoulook

up,theanglefromthehorizontaltoyourlineofsightiscalledtheangleofelevation.Likewise,if

youarelookingdown,theanglefromthehorizontaltoyourlineofsightiscalledtheangleof

depression.

7. Afterlookingatthisdescription,AndreamentionedthatherangleofelevationtoseeBonita

wasabout23.5°.Theybothagreed.BonitathensaidherangleofdepressiontoAndreawasabout66.5°.AndreaagreedthatitwasanangleofdepressionbutsaidBonita’sangleofdepressionwasalso23.5°.Whodoyouthinkiscorrect?Usedrawingsandwordstojustifyyourconclusion.

8. Whatconclusioncanyoumakeregardingtheangleofdepressionandtheangleofelevation?

Why?

Page 58






6.10


READY Topic:Modelingrealworldproblemswithtriangles.

Foreachstorypresentedbelowsketchapictureofthesituationandlabelasmuchofthepictureaspossible.Noneedtoanswerthequestionorfindthemissingvalues,simplyrepresentthesituationwithasketch.

1. Jillputaladderupagainstthehousetotryandreachalightthatisoutandneedstobechanged.Sheknowstheladderis10feetlongandthedistancefromthebaseofthehousetothebottomoftheladderis4feet.

2. Francisisapilotofanairplanethatifflyingatanaltitudeof3,000feetwhentheplanebeginsitsdescenttowardtheground.Iftheangleofdecentoftheplaneis150howmuchfartherwilltheplaneflybeforeitisontheground?

3. Abbyisstandingatthetopofaverytallskyscraperandlookingthroughatelescopeatthesceneryallaroundher.Theangleofdeclineonthetelescopesays350andAbbyknowssheis30floorsupandeachflooris15feettall.HowfarfromthebaseofthebuildingistheobjectthatAbbyislookingat?


Page 59






6.10


SETTopic:SolvingtrianglesusingTrigonometricRatios

Ineachtrianglefindthemissinganglesandsides.

4. 5.

m∠A = m∠B =

m∠C = 90o AC =

m∠A = AB =

m∠C = 90o BC =

6. 7.

AB = m∠B =

m∠C = 90o BC =

AB = m∠B =

m∠C = 90o BC =

Page 60






6.10


GO Topic:Trigonometricratios

Usethegivenrighttriangletoidentifythetrigonometricratios.Andangleswerepossible.

8.

sin(!) = cos(!) = tan(!) =

sin(!) = cos(!) = tan(!) =

9. sin(!) = cos(!) = tan(!) =

sin(!) = cos(!) = tan(!) =

m∠A= m∠B=

10.

sin(!) = cos(!) = tan(!) =

sin(!) = cos(!) = tan(!) =

m∠A= m∠B= m∠C=90o

Page 61

SECONDARY MATH III // MODULE 5

MODELING WITH GEOMETRY – 5.5


5.5 Special Rights


Inpreviouscoursesyouhavestudiedthe

Pythagoreantheoremandrighttriangletrigonometry.

Bothofthesemathematicaltoolsareusefulwhentryingtofindmissingsidesofarighttriangle.

1. WhatdoyouneedtoknowaboutarighttriangleinordertousethePythagorean

theorem?

2. Whatdoyouneedtoknowaboutarighttriangleinordertouserighttriangle

trigonometry?

WhileusingthePythagoreantheoremisfairlystraightforward(youonlyhavetokeeptrackof

thelegsandhypotenuseofthetriangle),righttriangletrigonometrygenerallyrequiresacalculator

tolookupvaluesofdifferenttrigratios.Therearesomerighttriangles,however,forwhich

knowingasidelengthandanangleisenoughtocalculatethevalueoftheothersideswithoutusing

trigonometry.Theseareknownasspecialrighttrianglesbecausetheirsidelengthscanbefoundby

relatingthemtoanothergeometricfigureforwhichweknowsomethingaboutitssides.

Onetypeofspecialrighttriangleisa45°-45°-90°triangle.

3. Drawa45°-45°-90°triangleandassignaspecificvaluetooneofitssides.(Forexample,letoneofthelegsmeasure5cm,orchoosetoletthehypotenusemeasure8inches.Youwill

wanttotrybothapproachestoperfectyourstrategy.)Nowthatyouhaveassigneda

measurementtooneofthesidesofyourtriangle,findawaytocalculatethemeasuresofthe

othertwosides.Aspartofyourstrategy,youmaywanttorelatethistriangletoanother

geometricfigurethatmaybeeasiertothinkabout.

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MODELING WITH GEOMETRY – 5.5


4. Generalizeyourstrategybylettingonesideofthetrianglemeasurex.Showhowthemeasureoftheothertwosidescanberepresentedintermsofx.(Makesuretoconsidercaseswherexisthelengthofaleg,aswellasthecasewherexisthelengthofthehypotenuse.)

Anothertypeofspecialrighttriangleisa30°-60°-90°triangle.

5. Drawa30°-60°-90°triangleandassignaspecificvaluetooneofitssides.Nowthatyouhaveassignedameasurementtooneofthesidesofyourtriangle,findawaytocalculatethe

measuresoftheothertwosides.Aspartofyourstrategy,youmaywanttorelatethis

triangletoanothergeometricfigurethatmaybeeasiertothinkabout.

6. Generalizeyourstrategybylettingonesideofthetrianglemeasurex.Showhowthemeasureoftheothertwosidescanberepresentedintermsofx.(Makesuretoconsidercaseswherexisthelengthofaleg,aswellasthecasewherexisthelengthofthehypotenuse.)

7. Canyouthinkofanyotheranglemeasurementsthatwillcreateaspecialrighttriangle?

Page 64


MODELING WITH GEOMETRY - 5.5

5.5



READY

Topic:Findingmissingmeasuresintriangles

Usethegivenfiguretoanswerthequestions.Roundyouranswerstothehundredthsplace.

Given:!∠$%& = ()°!∠$&+ = ,-°

1. Find.∠/01

Given:!∠$+& = 2-°

2. Find.∠/03456.∠301

Given:CA=6ft

3. FindBC

4. FindBA

5. FindCD

6. FindAD

7. FindBD

8. Findtheareaof∆/01

READY, SET, GO! Name PeriodDate

Page 65



5.5



SET

Topic:RecallingtrianglerelationshipsinSpecialRightTriangles

Fillinallthemissingmeasuresinthetriangles.

9. 10. 11.

12. 13. 14.

Useanappropriatetrianglefromabovetofillinthefunctionvaluesbelow.Nocalculators.

15.

sin45° =

cos 45° =

tan 45° =

16.

sin 30° =

cos 30° =

tan 30° =

17.

sin60° =

cos 60° =

tan 60° =

60° 30°

60°

60°

Page 66



5.5



GO Topic:Performingfunctionarithmeticonagraph

22. Writetheequationsoff(x)andg(x).

23. Writetheequationofthesumoff(x)andg(x).s(x)=

24. Writetheequationofthedifferenceoff(x)andg(x).d(x)=

25. Writetheequationoftheproductoff(x)andg(x).p(x)=

26. Writetheequationofthequotientoff(x)dividedbyg(x).

q(x)=

18. Addf(x)andg(x)usingthegraphattheright.

Drawthenewfigureonthegraphandlabelitass(x),thesumofx.

19. Subtractf(x)fromg(x)usingthegraphattheright.

Drawthenewfigureonthegraphandlabelitasd(x),thedifferenceofx.

20. Multiplyf(x)andg(x)onthesecondgraphattheright.

Drawthenewfigureonthegraphandlabelitasp(x),theproductofx.

21. Dividef(x)byg(x)onthesecondgraphattheright.

Drawthenewfigureonthegraphandlabelitasq(x),thequotientofx.

f(x)

f(x)

g(x)

g(x)

Page 67




6. 11 Solving Right TrianglesUsing Trigonometric Relationships


I. Foreachproblem:

• makeadrawing

• writeanequation

• solve(donotforgettoincludeunitsofmeasure)

1. Carrieplacesa10-footladderagainstawall.Iftheladdermakesanangleof65°withthelevelground,howfarupthewallisthetopoftheladder?

2. Aflagpolecastsashadowthat is15feet long.Theangleofelevationof thesunat thistimeis

40°.Howtallistheflagpole?

3. In southern California, there is a six-mile section of Interstate 5 that increases 2,500 feet in

elevation.Whatistheangleofelevation?

4. Ahotairballoonis100feetstraightabovewhereitisplanningtoland.Sarahisdrivingtomeet

theballoonwhenitlands.Iftheangleofelevationtotheballoonis35°,howfarawayisSarahfromplaceonthegroundwheretheballoonwillland?

5. An airplane is descending as it approaches the airport. If the angle of depression from the

plane to the ground is 7°, and theplane is 2,000 feet above the ground,what is thedistancefromtheplanetotheairport?

6. Michelleis60feetawayfromabuilding.Theangleofelevationtothetopofthebuildingis41°.Howtallisthebuilding?

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Page 69




7. Arampisusedforloadingequipmentfromadocktoaship.Therampis10feetlongandthe

shipis6feethigherthanthedock.Whatistheangleofelevationoftheramp?

II. Foreachrighttrianglebelow,findallunknownsidelengthsandanglemeasures:

8. 9.

10. 11.

12. Drawandfindthemissinganglemeasuresoftherighttrianglewhosesidesmeasure4,6,and8.

III. Determinethevaluesofthetworemainingtrigonometricratioswhengivenoneofthetrigonometricratios.

13. cos(!) = !!

14. tan ! = !!

15. sin(!) = !!

Page 70






6.11


READY Topic:Similartrianglesandproportionalrelationshipswithparallellines

Basedoneachsetoftrianglesorparallellinescreateaproportionandsolveittofindthemissing

values.

1. 2. 3.

4. 5. 6.


Page 71






6.11


SETTopic:SolvingtriangleswithtrigonometricratiosandPythagoreantheorem

Solveeachrighttriangle.Giveanymissingsidesandmissingangles.

7. 8.

9. 10.

11. 12.

Page 72






6.11


Usethegiventrigonometricratiotosketcharighttriangleandsolvethetriangle.

13. sin(!) = !! 14. cos(!)= !!

15. tan(!)= !! 16. sin(!)= !!"

17. cos(!)= !! 18. tan(!)= !!"

Usetherighttrianglebelowtodeterminewhichofthefollowingareequivalent.

19. sin(!) 20. cos(!)21. tan(!) 22. sin(!)23. cos(!) 24. tan(!)

25.!"#(!)!"#(!) 26.

!!"#(!)

27. 1 28. a2+b2

29. c2 30. !"#!(B)+!"#!(!)

Page 73






6.11


GO Topic:Applyingtrigonometricratiosandidentitiestosolveproblems.

Solveeachproblem.Sketchadrawingofthesituation.

31. Markisbuildinghissonapitcher’smoundsohecanpracticeforhisupcomingbaseballseasoninthebackyard.Markknowsthattheleaguerequiresaninclineof12°andanelevationof8inchesinheight.Howlongwillthefrontofthepitcher’smoundneedtobe?

32. Susanisdesigningawheelchairramp.Wheelchairrampsrequireaslopethatisnomorethan1-

inchofriseforevery12-inchesoframplength.Susanwantstodeterminehowmuchhorizontaldistancearampof6-feetinlengthwillspan?Shealsowantstoknowthedegreeofinclinefromthebaseofthe

ramptotheground.

33. Michaelisdesigningahousewitharoofpitchof5.Roofpitchisthenumberofinchesthataroof

willriseforevery12inchesofrun.Whatistheanglethatwillneedtobeusedinbuildingthetrusses

andsupportsfortheroof?Whatistheangleofaroofwith5/12pitchincrease?Atthepeakoftheroof

whatanglewilltherebewhenthefrontandthebackoftheroofcometogether?

Page 74

similarity & right triangle trigonometry...secondary math ii // module 6 similarity & right...

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