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Unit5ApplyingSimilarityofTriangles
Lesson1:ProofoftheTriangleSideSplitterTheoremOpeningExerciseWearegoingtoconstructaproofdesignedtodemonstratethefollowing:Alinesegmentparalleltoonesideofatriangledividestheothertwosidesproportionally.
a. Fillinthehypothesisandconclusion: If____________________________________________________________________________________, then_________________________________________________________________________________.b. Drawadiagram:c. Provethetheorem:d. Whatsimilaritytransformationmapsthesmallertriangleontothelargertriangle?
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Wecouldalsoprovetheconverse(switchthehypothesisandconclusion)oftheOpeningExerciseusingSASsimilarity.Thisgivesusthefollowingtheorem:
TriangleSideSplitterTheorem
Alinesegmentsplitstwosidesofatriangleproportionallyifandonlyifitisparalleltothethirdside.
Sonowbecauseofsimilartriangleswecanusetheproportion:DuetotheTriangleSideSplitterTheoremwecanusetheproportion:Toextendthisfurther,wecouldalsousetheproportion:Anyoftheseproportionscanbeusedtosolveanyprobleminvolvingthesidesofatrianglebeingsplitbyasegmentparalleltoathirdside.Typicallyonewaywillprovetobethemostefficientwayofsolving!!!
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Example1If , , ,and ,whatis ?Example2If , , ,and ,whatis ?
XY ! AC BX = 4 BA = 5 BY = 6 BC
XY ! AC BX = 9 BA = 15 BY = 15 YC
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ExercisesIn1-4,findxgiventhat DE ! AC .1. 2.
3 4.Example3Intheaccompanyingdiagram, ,DB=6andAE=8.IfECisthreetimesAD,findAD.
DE ! BC
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HomeworkIn1-3, andtriangleshavenotbeendrawntoscale.1. CB=8,BE=4andAD=3.FindAC. 2. CA=6,AD=2andBE=5.FindCE.
3. CA=8,BE=4.5andCBisfourtimesAD.FindCB.
AB ! DE
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Lesson2:ApplyingtheTriangleSideSplitterTheoremOpeningExerciseInthetwotrianglespicturedbelow, and .Findthemeasureofxinbothtriangles.WhatistherelationshipbetweenΔABC andΔFGH ?Sincethetwotrianglesshareacommonside,lookwhathappenswhenwepushthemtogether:Wenowhave3parallellinescutby3transversals.Isthetransversalontheleftinproportiontothetransversalontheright?
DE ! BC FG ! IJ
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Example1Wearegoingtodoaninformalproofofthefollowingtheorem:Theorem:If3ormoreparallellinesarecutby2transversals,thenthesegmentsofthetransversalsareinproportion.Bydrawinganauxiliarylinetocreatetwosimilartrianglesthatshareacommonside,wewillshow:
xy
= x 'y '
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Example2Inthediagrampicturedbelow, ,AB=20,CD=8,FD=12and .IftheperimeteroftrapezoidABDCis64,findAEandEC.
AB ! EF !CD AE :EC = 1: 3
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TheTriangleSideSplitterTheoremallowsustoworkwiththebasesofthetrianglesaswell.Sinceweknow ΔACB ∼ ΔDCE ,itsbesttolookatthesetrianglesseparately:
ACDC
= ABDE
or upper left sidewhole left side
= upper basewhole base
Wecannotusethelowersideswhenworkingwithbases!!!
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Exercises1. InthediagrambelowofΔABC , 2. InthediagrambelowofΔABC ,
MN ! AB ,AC=8,AB=12,and DE ! AC ,DB=6,AD=2,and CM=6.FindthelengthofMN. DE=9.FindAC.
3. Averticalpole,15feethigh,castsashadow12feetlong.Atthesametime,anearby
treecastsashadow40feetlong.Whatistheheightofthetree?4. Inthediagrampictured,alargeflagpolestandsoutsideofanofficebuilding.Josh
realizesthatwhenhelooksupfromtheground,60mawayfromtheflagpole,thatthetopoftheflagpoleandthetopofthebuildinglineup.Iftheflagpoleis35mtall,andJoshis170mfromthebuilding,howtallisthebuildingtothenearesttenth?
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Homework1. Findthevalueofx.Linesthatappear 2. If ,CA=4,AD=2
tobeparallelareinfactparallel. andAB=8.FindDE.
3. Inthediagrampicturedbelow, AB ! EF !CD ,AB=7,
CD=4,FD=6and AE :EC = 1: 4 .IftheperimeteroftrapezoidABDCis31,findAEandEC.
4. Kolbyneedstofixaleakyroofonhismom’shousebutdoesn’townaladder.He
thinksthata25-footladderwillbelongenoughtoreachtheroof,butheneedstobesurebeforehespendsthemoneytobuyone.HechoosesapointPonthegroundwherehecanvisuallyaligntheroofofhis4.25fttallcarwiththeedgeoftheroofofthehouse.IfpointPis8.5ftfromthecarandthecaris23ftfromthehouse,willthe25-footladderbetallenough?
AB ! DE
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Lesson3:PropertiesofSimilarTrianglesOpeningExerciseGiven ΔABC ∼ ΔA 'B 'C ' picturedtotheright:Findthelengthsofthemissingside.Findtheperimeterofthetriangles.Findtheareaofthetriangles.Example1Usingtheopeningexercise,answerthefollowing:a. Whatistherelationshipbetweenthesidesofthetrianglesandtheperimeters?b. Whatistherelationshipbetweenthesidesofthetrianglesandtheareas?c. Makeahypothesisoftheserelationships.
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Example2Giventhesimilarrectanglespicturedtotheright,find:a. Ratioofthesides b. Ratiooftheperimeters c. RatiooftheareasCONCLUSIONS:Whentwofiguresaresimilarandtheratiooftheirsidesisa:b,then:Theperimetersareintheratioof:Theareasareintheratioof:Ifthesimilarfigureswere3-dimensional,whatdoyouthinkistherelationshipbetweenthevolumesofthefigures?
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Exercises1. Twotrianglesaresimilar.Thesidesofthesmallertriangleare6,4,8.Iftheshortest
sideofthelargertriangleis6,findthelengthofthelongestside.2. Thesidesofatriangleare8,5,and7.Ifthelongestsideofasimilartriangle
measures24,findtheperimeterofthelargertriangle.3. Thesidesofatriangleare7,8and10.Whatisthelengthoftheshortestsideofa similartrianglewhoseperimeteris75?4. Findtheratiooftheareasoftwosimilartrianglesinwhichtheratioofthepairof
correspondingsidesis3:2.5. Findtheratioofthelengthsofapairofcorrespondingsidesintwosimilarpolygons
iftheratiooftheareasis4:25.
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6. Giventwosimilartriangles,theratiooftheareasis9:16.Theheightofthesmallertriangleis11inches,findtheheightofthelargertriangletothenearesttenth.
7. Caterina’sboathascomeuntiedandfloatedawayonthelake.Sheisstandingatopa
cliffthatis35feetabovethewaterinalake.Ifshestands10feetfromtheedgeofthecliff,shecanvisuallyalignthetopofthecliffwiththewateratthebackofherboat.Hereyelevelis5.5feetabovetheground.Howfaroutfromthecliff,tothenearesttenth,isCatarina’sboat?
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Homework1. Twotrianglesaresimilar.Thesidesofonetriangleare4,8and10.Iftheshortest
sideofthesecondtriangleis12,findthelengthofthemissingsides.2. Twotrianglesaresimilar.Thesidesofthesmallertriangleare6,4,and8.The
perimeterofthelargersimilartriangleis27.Findthelengthoftheshortestsideofthelargertriangle.
3. Findtheratiooftheareasoftwosimilartrianglesinwhichtheratioofthepairof
correspondingsidesis5:3.4. Findtheratioofthevolumesoftwosimilartrianglesinwhichtheratioofthepairof
correspondingareasis9:16.
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Lesson4:TheAngleBisectorTheoremOpeningExerciseTheAngleBisectorTheoremstates:
InΔABC ,iftheanglebisectorof∠A meetssideBCatpointD,thenBD:CD=BA:CA.
WearegoingtofillinthemissingreasonsoftheproofoftheAngleBisectorTheorem.Given:ADistheanglebisectorof∠A
Prove: BDCD
= BACA
Statements Reasons 1. ADistheanglebisectorof∠A 1.
2. ∠1≅ ∠2 2.
3. DrawCE ! toABwhereEisthepointof 3. intersectionofAD.(Label∠CED as∠3 )
4. ∠2 ≅ ∠3 4. 5. ∠4 ≅ ∠5 5.
6. ΔCDE ∼ ΔBDA 6.
7.BDCD
= BACE 7.
8. ∠1≅ ∠3 8.
9. ΔACE isisosceles 9.
10. CA ≅ CE 10.
11.BDCD
= BACA 11.
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5
4
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Exercises1. InΔABC picturedbelow,ADistheanglebisectorof∠A .IfCD=6,CA=8and
AB=12,findBD.2. InΔABC picturedbelow,ADistheanglebisectorof∠A .IfCD=9,CA=12and
AB=16,findBD.3. ThesidesofΔABC picturedbeloware10.5,16.5and9.Ananglebisectormeetsthe
sidelengthof9.Findthelengthsofxandy.
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Homework1. InΔABC pictured,ADistheangle
bisectorof∠A .IfBD=5,CD=6andAB=8,findAC.
2. TheconverseoftheAngleBisector
Theoremstates:
InΔABC ,ifAEmeetsBCatDsuchthatBD:CD=BA:CA,thenADistheanglebisectorof∠A .
Note:AB CE Statements Reasons
1. BDCD
= BACA 1. Given
2. ∠1≅ ∠2 2. Alt.interior∠ 's are≅
3. ∠3≅ ∠4 3. 4. 4.
5. 5.
6. CE ≅ CA 6. Thegivenandstep5implythis
7. isisosceles 7.
8. ∠1≅ ∠5 8.
9. ∠2 ≅ ∠5 9. Substitution
10. ADistheanglebisectorof 10.
!
ΔCDE ∼ ΔBDA
BDCD
= BACE
ΔACE
∠A
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Lesson5:SpecialRelationshipsWithinRightTrianglesOpeningExerciseGiventhetrianglespicturedabove,answerthefollowing:a. Arethetrianglessimilar?Explain.b. Findallofthemissingsidesandangles.
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Example1In picturedtotheright,∠B isarightangleandBDisthealtitude.a. Howmanytrianglesdoyouseeinthepicture?b. Identifythetrianglesbyname.c. Isthebigtrianglesimilartothesmalltriangle?Explain.d. Isthebigtrianglesimilartothemediumtriangle?Explain.e. Whatistherelationshipbetweenthesmalltriangleandthemediumtriangle?How
doyouknow?
ΔABC
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Lesson6:SimplifyingRadicalsOpeningExercisea. Writetheproductsincolumn2.
Factors Products1 ⋅ 1or12
2 ⋅2or 22
3 ⋅3or 32
4 ⋅4or
5 ⋅5or
6 ⋅6or
7 ⋅7or
8 ⋅8or
9 ⋅9or
10 ⋅ 10or
11 ⋅ 11or
12 ⋅ 12or
13 ⋅ 13or
14 ⋅ 14or
15 ⋅ 15or
b. WhatisanothernamefortheproductsinColumn2? c. Nowlet’slookatvariables:
Factors Products
x ⋅ x
x2 ⋅ x2
x3 ⋅ x3
x4 ⋅ x4
x5 ⋅ x5
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Example1 Simplifythesquarerootofeachperfectsquarefromcolumn1.
Perfect Squares SimplifiedAnswer
1
4
9
16
25
36
49
64
81
100
121
144
Simplifyeachperfectsquarevariable:
Perfect Squares SimplifiedAnswer
x2
x4
x6
x8
Simplifythefollowing:
a. 64x2 b. 2500x1000 c.4x4
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Example2SimplifyingNON-PERFECTsquares:
Non-Perfect Square =
Largest perfect square • remaining factor
Simplifythefollowing:
a. 32 b. 500 c. 16x3 d. 3600x5 e. 48x3 f. 125 g. 3 45x2 h. 2 52 j. 16 16
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Lesson7:MultiplyingandDividingRadicalsOpeningExerciseSimplifyeachofthefollowing:
3600x500 25x3 98 Example1
MultiplyingandDividingRadicals:
1. Multiply/DividetheCoefficient2. Multiply/DividetheRadicand(numberundertheradical)3. Simplify
Simplifythefollowing:
a. ( 6 ) ⋅ ( 60) b. 964
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ExercisesSimplify:
1. 1725 2. ( x )( 4x )
3.
€
4 30( ) 6 3( ) 4.
€
16 202 5
RationalizingDenominators(youareNOTallowedtohavearadicalinthedenominator)
1. Checktoseeifyoucandividethenumeratoranddenominatortoeliminate
theradical.2. IfyouCANNOTeliminatetheradicalinthedenominator,youmust
rationalizethedenominator.3. Multiplethenumeratoranddenominatorbytheradicalinthedenominator.4. Simplifythenumeratoranddenominator.
Example2Rationalizethefollowing:
1. 12 2. 7
5
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Exercises1. Findtheareaoftherectangleinsimplestradical
form.2. Findthewidthoftherectanglewithanareaof 4 6
insimplestradicalform.
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HomeworkIn1-6,simplifythegivenexpressions:1. 52 2. 48x6 3.
€
5 3( ) 7 2( ) 4.
€
3 2( ) 14( )
5.
€
10 65 2
6.
€
23
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Lesson8:AddingandSubtractingRadicalsOpeningExerciseCalculatetheareaofthetriangle: Thetriangleshownbelowhasaperimeterof6.5 2 units.Makeaconjectureabouthowthisanswerwasreached.
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Example1 Simplifythefollowingexpressions:a. 3 18 +10 2 b. 19 2 −2 8
AddingandSubtractingRadicals:
1. Simplifyallterms2. Add/subtractthecoefficientsoflikeradicals3. Keeptheradical
Exercises: Simplifythefollowingexpressions:1. 18 5 −12 5 2. 24 + 5 24 3. 2 7 + 4 63
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4. Whatistheperimeterofthetriangleshownbelow?
5. Determinetheareaofthetriangleshown.Simplifyasmuchaspossible.
6. Determinetheareaandperimeteroftherectangleshown.Simplifyasmuchas
possible
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HomeworkSimplifythefollowingexpressions:1.
€
5 28 2.
€
2 3 + 5 3
3.
€
8 + 18 − 2 4.
€
50 + 986
5. Determinetheareaandperimeterofthetriangleshown.Simplifyasmuchas
possible.
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Lesson9: ProofofPythagoreanTheoremOpeningExerciseInthetrianglepicturedbelow,findxandy:
*****Thereisanothermethodthatcanbeusedtofindxandy.******
PythagoreanTheorem:
a2 + b2 = c2
Usetofindthemissingsideofarighttrianglewhen2sidesaregiven.
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Example1WearegoingtoprovethePythagoreanTheorem!Usethefollowingpicturetoseparateoutthe“small”,“medium”and“large”triangles.Labeleachtrianglewiththeappropriatevertexandsidelabels. “small” “large” “medium”BasedonwhatwedidinLesson6,weknowthatall3trianglesaresimilar,sowecansetupproportions.a. Whichtwotrianglesshare∠B ?b. Writeaproportionthatcomparesthesetwotrianglesusingtheratio longerleg:hypotenuse,thenperformcrossproducts.c. Whattwotrianglesshare∠A ?d. Writeaproportionthatcomparesthesetwotrianglesusingtheratio shorterleg:hypotenuse,thenperformcrossproducts.e. Ourgoalistoshowthata2 + b2 = c2 .Let’susesubstitutionandourcrossproducts.
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Exercises1. Findthemissingsides,tothenearesttenth: a. b. 2. Tanyarunsdiagonallyacrossarectangularfieldthathasalengthof40yardsanda
widthof30yards.Whatisthelengthofthediagonal,inyards,thatTanyaruns?3. Thesidesofatriangleare7,11,and18.Dothesesidesformarighttriangle?Justify
youranswer.