5.7 Justifying the Laws - Utah Education Network · MODELING WITH GEOMETRY – 5.7 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org

SECONDARY MATH III // MODULE 5

MODELING WITH GEOMETRY – 5.7

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

5.7 Justifying the Laws

A Solidify Understanding Task

ThePythagoreantheoremmakesaclaimabout

therelationshipbetweentheareasofthethree

squaresdrawnonthesidesofarighttriangle:thesumoftheareaofthesquaresonthetwolegsis

equaltotheareaofthesquareonthehypotenuse.Wegenerallystatethisrelationship

algebraicallyas!" + $" = &" ,whereitisunderstoodthataandbrepresentthelengthofthetwolegsoftherighttriangle,andcrepresentsthelengthofthehypotenuse.

Whataboutnon-righttriangles?Istherearelationshipbetweentheareasofthesquares

drawnonthesidesofanon-righttriangle?(Note:ThefollowingproofisbasedonTheIllustratedLaw

ofCosines,byDonMcConnellhttp://www.cut-the-knot.org/pythagoras/DonMcConnell.shtml)

Thediagramonthenextpageshowsanacutetrianglewithsquaresdrawnoneachofthethree

sides.Thethreealtitudesofthetrianglehavebeendrawnandextendedthroughthesquaresonthe

sidesofthetriangle.Thealtitudesdivideeachsquareintotwosmallerrectangles.

1. Findanexpressionfortheareasofeachofthesixsmallrectanglesformedbythealtitudes.Writetheseexpressionsinsideeachrectangleonthediagram.(Hint:Theareaofeachrectanglecanbeexpressedastheproductofthesidelengthofthesquareandthelengthofasegmentthatisalegofarighttriangle.Youcanuserighttriangletrigonometrytoexpressthelengthofthissegment.)

2. Althoughnoneofthesixrectanglesarecongruent,therearethreepairsofrectangleswhereeachrectangleinthepairhasthesamearea.Usingthreedifferentcolors—red,blueandgreen—shadepairsofrectanglesthathavethesameareawiththesamecolor.

3. Theareaofeachsquareiscomposedoftwosmaller,rectangularareasoftwodifferentcolors.Writethreedifferent“equations”torepresenttheareasofeachofthesquares.Forexample,youmightwrite!" = $'() + *)+ifthosearethecolorsyouchosefortheareasoftherectanglesformedinthesquaredrawnonsidea.

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4. Selectoneofyourequationsfromstep3,suchas ,andusetheothertwosquarestosubstituteadifferentexpressioninforeachcolor.Forexample,ifinyourdiagram

�

a2 = blue + red

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and ,wecanwritethisequation:

or .

Writeyourselectedequationinitsmodifiedformhere:

5. Sinceeachcolorisactuallyavariablerepresentinganareaofarectangle,replacetheremainingcolorinyourlastequationwiththeexpressionthatgivestheareaoftherectanglesofthatcolor.

Writeyourfinalequationhere:

6. Repeatsteps4and5fortheothertwoequationsyouwroteinstep3.YoushouldendupwiththreedifferentversionsoftheLawofCosines,eachrelatingtheareaofoneofthesquaresdrawnonasideofthetriangletotheareasofthesquaresontheothertwosides.

7. WhathappenstothisdiagramifangleCisarightangle?(Hint:Thinkaboutthealtitudesinarighttriangle.)

8. Whydowehavetosubtractsomeareafrom toget whenangleCislessthanright?

�

blue = b2 − green

�

red = c 2 − green

�

a2 = b2 − green + c 2 − green

�

a2 = b2 + c 2 − 2 ⋅ green

�

a2 =

b2 =

c 2 =

�

a2 + b2

�

c 2

37




TheLawofCosinescanalsobederivedforanobtusetrianglebyusingthealtitudeofthe

triangledrawnfromthevertexoftheobtuseangle,asinthefollowingdiagram,whereweassumethat

angleAisobtuse.

9. UsethisdiagramtoderiveoneoftheformsoftheLawofCosinesyouwroteabove.(Hint:Asintheprevioustask,MoreThanRight,thelengthofthealtitudecanberepresentedintwodifferentways,bothusingthePythagoreantheoremandtheportionsofsideathatformthelegsoftwodifferentrighttriangles.)

10. UsethesamediagramabovetoderivetheLawofSines.(Hint:Howcanyourepresentthelengthofthealtitudeintwodifferentwaysusingsidesa,b,orcandrighttriangletrigonometryinsteadofthePythagoreantheorem?)

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5.7 Justifying the Laws – Teacher Notes A Solidify Understanding Task

Purpose:InthistaskstudentsexamineproofsoftheLawofCosinesandtheLawofSinesusingthe

geometricandalgebraicstrategiesdevelopedintheprevioustask.Generictrianglesinthistaskare

labeledusingtheconventionthatthesideoppositeÐAislabeledassidea,thesideoppositeÐBis

labeledassideb,andthesideoppositeÐCislabeledassidec.Studentsexaminehowtheycan

representthelengthofalegofarighttriangleABCwithatrigexpressionintermsofalabeledangle

andalabeledside(e.g.,“bcos(A)”).SuchexpressionsareusedinthealgebraicderivationoftheLawof

CosinesandLawofSines.Thenexttaskwillprovidestudentswithopportunitiestopracticetheselaws

inapplications,suchasfindingtheareaofatriangle.

Notetoteachers:Sincestudentshaveonlyworkedwithrighttriangletrigonometry,findingthesine

orcosineofananglemeasuringgreaterthan90hasnomeaning,sincesuchanglesdonotexistinrighttriangles.Thistask,andtheRSGhomeworkthataccompaniesthistask,takesthisrestrictioninto

account.

CoreStandardsFocus:

G.SRT.10(+)ProvetheLawsofSinesandCosinesandusethemtosolveproblems.

G.SRT.11(+)UnderstandandapplytheLawofSinesandtheLawofCosinestofindunknown

measurementsinrightandnon-righttriangles(e.g.,surveyingproblems,resultantforces).

StandardsforMathematicalPractice:

SMP7–Lookforandmakeuseofstructure




TheTeachingCycle:

Launch(WholeClass):

Studentswillneedtounderstandhowthediagramreferredtoinquestions1-6wasconstructedso

theycanmakesenseofthealgebraicworkofthesequestions.IfyouhaveGeometer’sSketchpador

similardynamicgeometrysoftware,youmaywanttohavestudentsconstructthisdiagramusingthe

software,orprovideapre-constructedversionofthediagramforthestudentstouse.Ifstudentsare

constructingthisdiagramforthemselves,itwillhelptofirstcreateacustom“square”toolsothethree

squaresonthesidesofthetriangleareeasytocreate.Whetherornotyouareusingdynamic

geometrysoftwareorthestaticdiagramonthesecondpageofthetask,helpstudentsunderstandhow

thediagramwascreated:first,anacutetriangleABCwasconstructedusingthreearbitrarypointsas

vertices;second,asquarewasconstructedoneachsideofthetriangle;third,thelinescontainingthe

threealtitudesofthetrianglewereconstructed—theselinesdividingeachsquareintotwosmaller

rectangles.

Noteforstudents:WhenwerefertoÐA,ÐBorÐCwearereferringtotheanglesoftheoriginal

triangle,eventhoughthealtitudesformadditionalanglesateachvertex.Also,thetrianglehasbeen

labeledinthestandardway,withthesideoppositeÐAlabeledassidea,etc.

Modelthealgebraicworkofquestion1byfindinganexpressionforoneofthesixsmallrectangles.

Forexample,ifthesegmenthighlightedinthefollowingdiagramhasunknownlengthx,then

,so ,andtheareaoftheshadedrectangleis .Askstudentstofind

similarwaystolabelallsixrectangles,andthenhavethemcontinuewithquestions2-6.

�

cosC =xa

�

x = acosC

�

b ⋅ acosC




Explore(SmallGroup):

Forquestion2studentsshouldendupwithacoloreddiagram,similartothefollowing.(Note:

Becausestudentsarechoosingwhichcolortouseforeachareaexpression,theirdiagramsmaybe

coloreddifferentlyfromwhatisshownbelow;however,thesamepairsofrectanglesshouldsharethe

samecolors.)

Basedonthecolor-codingofthediagrambelow,forquestion3studentswouldwrite:

�

a2 = blue + redb2 = blue + greenc 2 = green + red




Usingthediagramandthesuggestionsgivenin4and5,studentsshouldbeabletoderivethethree

formsoftheLawofCosines:

Question7isintendedtohelpstudentsnoticethatthePythagoreantheoremisaspecialcaseofthe

LawofCosineswithrighttriangles.IfÐCisarightangle,thenthelegsarethealtitudes,sothesquares

onthetwolegsdonotgetdividedintosmallerrectangles.Onlythesquareonthehypotenusewillbe

�

a2 = b2 + c 2 − 2bc cosAb2 = a2 + c 2 − 2ac cosBc 2 = a2 + b2 − 2abcosC




dividedintotwosmallerrectanglesbythealtitudedrawnfromC.Thesetworectanglesformedonthe

hypotenusewillbethesamecolorsasthetwosquaresontheothertwosides.Thisisbestillustrated

byusingadynamicsketchofthediagramanddraggingpointCuntilitformsarightangle.

Thisderivationfallsapartifoneoftheanglesisobtuse.Therefore,question9asksstudentstoderive

theLawofCosinesintermsofadiagramthatdoesworkforobtusetriangles.Thissamediagramis

usedinquestion10toderivetheLawofSines.

FortheLawofCosines,studentswillneedtousethedecompositionofsideaintotwosmaller

segmentsoflengthxanda–x.Theyshouldignorethelabelsonthesetwosmallersegmentswhen

derivingtheLawofSines.(Thiswillfocustheirattentiononusingthesineratioratherthanthe

cosineratio.Ifstudentsareusingthecosineratioforquestion10,pointoutthatintheLawofCosines

weusedthecosineratio,buttheintentoftheLawofSinesistofindarelationshipbetweenthesinesof

theangles.ThisderivationoftheLawofSineswillinvolvetheratiossinBandsinC.)

Discuss(WholeClass):

Ifneeded,haveastudentpresenthowtheyderivedtheLawofCosinesusingthecolor-coded

rectangles.

FocustheremainderofthediscussiononderivingtheLawofCosinesandtheLawofSinesusingthe

diagramgivenpriortoquestion9.Pointoutthatthisdiagramworksequallywellforobtuseandacute

triangles.Ifavailable,usestudentworktooutlinethisproof.Ifnecessary,prompttheworkofthe

derivationbyaskingquestionssuchas,“Howcouldwedeterminethelengthofsegmenthintwo

differentways?




DerivationoftheLawofCosinesbasedonthisdiagram:

IfyouhadtodoalotofpromptingonthederivationoftheLawofCosines,givestudentsafewextra

minutestoworkonderivingtheLawofSinesforthemselves,sinceitinvolvessimilarreasoning.

DerivationoftheLawofSinesbasedonthisdiagram:

Thislaststatementisequivalentto ,themoreconventionalformforwritingtheLawof

Sines.

PointouttostudentsthatifDABCisacutewecoulduseanaltitudedrawnfromangleBorangleCto

showthattheratio isalsoequivalentto or ,leadingtothemoreextendedversionof

theLawofSines, .(Thiscanalsobeshowntobetrueforobtusetriangles,butit

requiresthatstudentsworkwithanaltitudethatliesoutsideofthetriangle,andtheuseofatrig

identity,sinA=sin(180°–A),whichstudentsdonotyethaveaccessto.Seeteachernoteabove.)

AlignedReady,Set,Go:ModelingwithGeometry5.7

�

x 2 + h2 = c 2 ⇒ h2 = c 2 − x 2

(a − x)2 + h2 = b2 ⇒ h2 = b2 − (a − x)2 = b2 − (a2 − 2ax + x 2) = b2 − a2 + 2ax − x 2

b2 − a2 + 2ax − x 2 = c 2 − x 2

b2 = a2 + c 2 − 2ax

cos B =xc

⇒ x = c cosB

b2 = a2 + c 2 − 2ac cosB

�

sinB =hc

⇒ h = c sinB

sinC =hb

⇒ h = bsinC

c sinB = bsinC

�

sinBb

=sinCc

�

sinAa

�

sinBb

�

sinCc

�

sinAa

=sinBb

=sinCc


MODELING WITH GEOMETRY - 5.7

5.7

Needhelp?Visitwww.rsgsupport.org


READY Topic:RecallingcircumferenceandareaofacircleUsethegiveninformationtofindtheindicatedvalue.Leaveπinyouranswer.Includethecorrectunit.

1.radius=3ft

circumference:

area:

2.diameter=14cm

circumference:

area:

3.circumference=38πkm

radius:

area:

4.area=49πin2

diameter:

circumference:

5.circumference=15πmi

radius:

area:

6.area=121πm2

radius:

circumference:

Solveforthemissingangle.Roundyouranswerstothenearestdegree.

(Hint:Inproblems10,11,and12,getthetrigfunctionalone.Thensolvefor!. )

7.cos ! = )*

8.tan ! = ./ 9.sin ! = 1

2

10.5 sin ! − 2 = 0

11.7 cos ! − 6 = 0 12.4 tan ! − 1 = 0

SET Topic:UsingtheLawsofsineandcosinetosolvetriangles

LawofSines:IfABCisatrianglewithsidesa,

b,andc,then ;<=>?

= @<=>A

= B<=>C

oritcanbewrittenas:

sinDE

= sinFG

=sin HI

LawofCosines:IfABCisatrianglewithsidesa,b,andc,then

E. = G. +I. − 2GI cos DG. = E. +I. − 2EI cos FI. = E. +G. − 2EG cos H

READY, SET, GO! Name PeriodDate

39



5.7



UsetheLawofsinestosolveeachtriangle.

13.

14.

15.

16.

17.WhatinformationdoyouneedinordertousetheLawofsines?

18.UsetheLawofcosinestofindtheremaininganglesandsideofthetriangle.

19.UsetheLawofcosinestofindtheremaininganglesandsideofthetriangle.

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5.7



20.UsetheLawofcosinestofindthethree

anglesofthetriangle.

21.UsetheLawofcosinestofindthethreeanglesofthetriangle.

22.WhatinformationdoyouneedinordertousetheLawofcosinestosolveatriangle?

GO Topic:RecallinghetrigratiosofthespecialrighttrianglesFillinthemissingangle.DoNOTuseacalculator.

23.sin ! = √..

24.tan ! = √3

25.cos ! = ).

26.sin ! = √/.

27.tan ! = 1

28.tan ! = √//

29.sin ! = ).

30.cos ! = √..

31.cos ! = √/.

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5.7 Justifying the Laws - Utah Education Network · MODELING WITH GEOMETRY – 5.7 Mathematics Vision Project Licensed under the Creative Commons Attribution CC BY 4.0 mathematicsvisionproject.org