1

Unit5ApplyingSimilarityofTriangles

Lesson1:ProofoftheTriangleSideSplitterTheoremOpeningExerciseWearegoingtoconstructaproofdesignedtodemonstratethefollowing:Alinesegmentparalleltoonesideofatriangledividestheothertwosidesproportionally.

a. Fillinthehypothesisandconclusion: If____________________________________________________________________________________, then_________________________________________________________________________________.b. Drawadiagram:c. Provethetheorem:d. Whatsimilaritytransformationmapsthesmallertriangleontothelargertriangle?

2

Wecouldalsoprovetheconverse(switchthehypothesisandconclusion)oftheOpeningExerciseusingSASsimilarity.Thisgivesusthefollowingtheorem:

TriangleSideSplitterTheorem

Alinesegmentsplitstwosidesofatriangleproportionallyifandonlyifitisparalleltothethirdside.

Sonowbecauseofsimilartriangleswecanusetheproportion:DuetotheTriangleSideSplitterTheoremwecanusetheproportion:Toextendthisfurther,wecouldalsousetheproportion:Anyoftheseproportionscanbeusedtosolveanyprobleminvolvingthesidesofatrianglebeingsplitbyasegmentparalleltoathirdside.Typicallyonewaywillprovetobethemostefficientwayofsolving!!!

3

Example1If , , ,and ,whatis ?Example2If , , ,and ,whatis ?

XY ! AC BX = 4 BA = 5 BY = 6 BC

XY ! AC BX = 9 BA = 15 BY = 15 YC

4

ExercisesIn1-4,findxgiventhat DE ! AC .1. 2.

3 4.Example3Intheaccompanyingdiagram, ,DB=6andAE=8.IfECisthreetimesAD,findAD.

DE ! BC

5

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

6

Lesson2:ApplyingtheTriangleSideSplitterTheoremOpeningExerciseInthetwotrianglespicturedbelow, and .Findthemeasureofxinbothtriangles.WhatistherelationshipbetweenΔABC andΔFGH ?Sincethetwotrianglesshareacommonside,lookwhathappenswhenwepushthemtogether:Wenowhave3parallellinescutby3transversals.Isthetransversalontheleftinproportiontothetransversalontheright?

DE ! BC FG ! IJ

7

Example1Wearegoingtodoaninformalproofofthefollowingtheorem:Theorem:If3ormoreparallellinesarecutby2transversals,thenthesegmentsofthetransversalsareinproportion.Bydrawinganauxiliarylinetocreatetwosimilartrianglesthatshareacommonside,wewillshow:

xy

= x 'y '

8

ExercisesInexercises1and2,findthevalueofx.Linesthatappeartobeparallelareinfactparallel.1. 2.

9

Example2Inthediagrampicturedbelow, ,AB=20,CD=8,FD=12and .IftheperimeteroftrapezoidABDCis64,findAEandEC.

AB ! EF !CD AE :EC = 1: 3

10

TheTriangleSideSplitterTheoremallowsustoworkwiththebasesofthetrianglesaswell.Sinceweknow ΔACB ∼ ΔDCE ,itsbesttolookatthesetrianglesseparately:

ACDC

= ABDE

or upper left sidewhole left side

= upper basewhole base

Wecannotusethelowersideswhenworkingwithbases!!!

11

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?

12

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

13

Lesson3:PropertiesofSimilarTrianglesOpeningExerciseGiven ΔABC ∼ ΔA 'B 'C ' picturedtotheright:Findthelengthsofthemissingside.Findtheperimeterofthetriangles.Findtheareaofthetriangles.Example1Usingtheopeningexercise,answerthefollowing:a. Whatistherelationshipbetweenthesidesofthetrianglesandtheperimeters?b. Whatistherelationshipbetweenthesidesofthetrianglesandtheareas?c. Makeahypothesisoftheserelationships.

14

Example2Giventhesimilarrectanglespicturedtotheright,find:a. Ratioofthesides b. Ratiooftheperimeters c. RatiooftheareasCONCLUSIONS:Whentwofiguresaresimilarandtheratiooftheirsidesisa:b,then:Theperimetersareintheratioof:Theareasareintheratioof:Ifthesimilarfigureswere3-dimensional,whatdoyouthinkistherelationshipbetweenthevolumesofthefigures?

15

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.

16

6. Giventwosimilartriangles,theratiooftheareasis9:16.Theheightofthesmallertriangleis11inches,findtheheightofthelargertriangletothenearesttenth.

7. Caterina’sboathascomeuntiedandfloatedawayonthelake.Sheisstandingatopa

cliffthatis35feetabovethewaterinalake.Ifshestands10feetfromtheedgeofthecliff,shecanvisuallyalignthetopofthecliffwiththewateratthebackofherboat.Hereyelevelis5.5feetabovetheground.Howfaroutfromthecliff,tothenearesttenth,isCatarina’sboat?

17

Homework1. Twotrianglesaresimilar.Thesidesofonetriangleare4,8and10.Iftheshortest

sideofthesecondtriangleis12,findthelengthofthemissingsides.2. Twotrianglesaresimilar.Thesidesofthesmallertriangleare6,4,and8.The

perimeterofthelargersimilartriangleis27.Findthelengthoftheshortestsideofthelargertriangle.

3. Findtheratiooftheareasoftwosimilartrianglesinwhichtheratioofthepairof

correspondingsidesis5:3.4. Findtheratioofthevolumesoftwosimilartrianglesinwhichtheratioofthepairof

correspondingareasis9:16.

18

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.

12

5

4

19

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.

20

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

21

Lesson5:SpecialRelationshipsWithinRightTrianglesOpeningExerciseGiventhetrianglespicturedabove,answerthefollowing:a. Arethetrianglessimilar?Explain.b. Findallofthemissingsidesandangles.

22

Example1In picturedtotheright,∠B isarightangleandBDisthealtitude.a. Howmanytrianglesdoyouseeinthepicture?b. Identifythetrianglesbyname.c. Isthebigtrianglesimilartothesmalltriangle?Explain.d. Isthebigtrianglesimilartothemediumtriangle?Explain.e. Whatistherelationshipbetweenthesmalltriangleandthemediumtriangle?How

doyouknow?

ΔABC

23

ExercisesFindthevalueofx:1. 2. 3. 4.5. 6.

24

Homework1. 2.3. 4.

25

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

26

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

27

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

28

HomeworkIn1-5,simplifythegivenexpressions:1. 144 2. 3 72 3. 3 32x2 4. 4 16x3 5. 1000x1000

29

Lesson7:MultiplyingandDividingRadicalsOpeningExerciseSimplifyeachofthefollowing:

3600x500 25x3 98 Example1

MultiplyingandDividingRadicals:

1. Multiply/DividetheCoefficient2. Multiply/DividetheRadicand(numberundertheradical)3. Simplify

Simplifythefollowing:

a. ( 6 ) ⋅ ( 60) b. 964

30

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

31

Exercises1. Findtheareaoftherectangleinsimplestradical

form.2. Findthewidthoftherectanglewithanareaof 4 6

insimplestradicalform.

32

HomeworkIn1-6,simplifythegivenexpressions:1. 52 2. 48x6 3.

€

5 3( ) 7 2( ) 4.

€

3 2( ) 14( )

5.

€

10 65 2

6.

€

23

33

Lesson8:AddingandSubtractingRadicalsOpeningExerciseCalculatetheareaofthetriangle: Thetriangleshownbelowhasaperimeterof6.5 2 units.Makeaconjectureabouthowthisanswerwasreached.

34

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

35

4. Whatistheperimeterofthetriangleshownbelow?

5. Determinetheareaofthetriangleshown.Simplifyasmuchaspossible.

6. Determinetheareaandperimeteroftherectangleshown.Simplifyasmuchas

possible

36

HomeworkSimplifythefollowingexpressions:1.

€

5 28 2.

€

2 3 + 5 3

3.

€

8 + 18 − 2 4.

€

50 + 986

5. Determinetheareaandperimeterofthetriangleshown.Simplifyasmuchas

possible.

37

Lesson9: ProofofPythagoreanTheoremOpeningExerciseInthetrianglepicturedbelow,findxandy:

*****Thereisanothermethodthatcanbeusedtofindxandy.******

PythagoreanTheorem:

a2 + b2 = c2

Usetofindthemissingsideofarighttrianglewhen2sidesaregiven.

38

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.

39

Example2 Let’sgobacktotheopeningexercisetofindxandyusingPythagoreanTheorem.

40

Exercises1. Findthemissingsides,tothenearesttenth: a. b. 2. Tanyarunsdiagonallyacrossarectangularfieldthathasalengthof40yardsanda

widthof30yards.Whatisthelengthofthediagonal,inyards,thatTanyaruns?3. Thesidesofatriangleare7,11,and18.Dothesesidesformarighttriangle?Justify

youranswer.

41

Homework1. Findthemissingsides: a. b.2. A12-footladderleansagainstahouse.Thefootoftheladderis7feetawayfromthe

house.Howhighupthehousedoestheladderreach,tothenearesttenth?3. Atrianglehassides20,21and29.Isitarighttriangle?Justifyyouranswer.