BigIdeasCh9Notes

Geometry Name___________________________________________ Date____________________________Period__________

Chapter9RightTrianglesandTrigonometry

9.1ThePythagoreanTheorem

Pythagorean Theorem

Converse of the Pythagorean

Theorem

Pythagorean Inequalities Theorem

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9.2SpecialRightTriangles

AreaExample:Theroadsignisshapedlikean___________________________________.Es?matetheareaofthesign.

Triangle Inequality Theorem (Ch 6)

45°- 45°- 90° Triangle Theorem

30°- 60°- 90° Triangle Theorem

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9.3SimilarRightTriangles

Createasimilaritystatementforthethreetriangles∆ABC,∆ACD,and∆BCD.

∆ABC~∆_______________~∆_______________

Iden?fythesimilartrianglesinthefigureontherightandcreatetheirsimilaritystatements.

Calcula@ngMean

The____________________________________betweentwonumbersrandsisdefinedtobe� .

The____________________________________xbetweentwonumbersrandsisdefinedtosa?sfythefollowingexpressions:

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Example:TheGeometricMeanbetween12and24is:

GeometricMeanandRightTrianglesLet’susethesimilaritystatementofthethreetriangles△CBD~△ACD~△ABC.Remember,correspondingsidesarepropor?onal.

Computetheunknownvalues.

Right Triangle Similarity Theorem

r + s2

xr= sx

x 2 =

Geometric Mean (Altitude) Theorem

Geometric Mean (Leg) Theorem

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Example:Asurveyor’slineofsighttothetopofacliffandhislineofsighttotheboVomformarightangle.Whatistheheightoftheclifftothenearestfoot?

9.4TheTangentRa@o

RightTriangles:Arethesetrianglessimilar?

TrigonometricRa@o:Thera?oof____________________________________________________

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TangentRa@o:Thera?oof____________________________________________________

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Example:Findthetangentra?os.

Example:Calculatexforeachtriangle.

AngleofEleva@on:Theangle_____________________________________________________

Computetheheightofthetreepicturedontheright.

tan 38˚( ) =

tan ∠A( ) =

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9.5TheSineandCosineRa@os

SineandCosineRa@os:Thera?oofa__________________and____________________________ofarighttriangle.

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Example:Findthesineandcosinera?osoftherighttriangle.

SineandCosineofComplementaryAngles

Thesineofan____________________angleisequaltothecosineofits________________________.

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UsingTrigonometricExpressionsa)Writesin(56˚)intermsofcosine. b)Usesineandcosinetocalculatexandy.

sin(56˚)=cos(_______________)

SpecialRightTriangles

AngleofDepression:___________________________________________________

Computethedistancexskiing.

ALiJleStory

Chief_________________________________

sin∠A = cos∠A =

sin A = cos 90˚−A( ) = cosB cos A =

sinB = cosB =

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9.6SolvingRightTriangles

InverseTrigonometricFunc@ons

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Calcula@ngAnglesComputetheacuteanglesA,B,andC.

a)tanA= b)sinB= c)cosC=

SolvetheTriangleComputesidecandanglesAandB. Computeg,h,andangleG.

RealWorldProblemYourschoolisbuildingarakedstage.

LondonEyeProblemTheobserva?onwheelhasaradiusof67.5mandtakes30minutestomakeacompleterota?on.

sin−1 sin∠A( ) = cos−1 cos∠A( ) =

tan−1 tan∠A( ) =

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9.7LawofSinesandLawofCosines

Calcula@ngObtuseAnglesUseyourcalculatortocomputetheseanglesandno?cethevalues.

a)tan_______= b)sin_______= c)cos_______=

AreaofaTriangle

Heron’sformula: Sineformula:

ExampleCalculatetheareaofthetriangle.

Solveforalltheunknownsidelengthsandangles.SolvetheTriangle(SSA) SolvetheTriangle(AAS)

SolvetheTriangle(ASA)

Law of Sines

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LawofSinesisbestfortrianglesthathave:___________________,___________________,___________________

LawofCosinesisbestfortrianglesthathave:___________________,___________________

SolvetheTriangle(SAS) SolvetheTriangle(SSS)

Law of Cosines

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