handbook of geometry

8/13/2019 Handbook of Geometry

1/82

Copyright 2010-2013, Earl Whitney, Reno NV. All Rights Reserved

Math Handbook

of Formulas, Processes and Tricks

Geometry

Prepared by: Earl L. Whitney, FSA, MAAA

Version 2.2

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GeometryHandbook

Page Description

Chapter1:Basics

a eo ontents

,

7 Segments,Rays&Lines

8 DistanceBetweenPoints(1Dimensional,2Dimensional)

9 DistanceFormulainn Dimensions

10 Angles

11 TypesofAngles

ap er : roo s

12 ConditionalStatements(Original,Converse,Inverse,Contrapositive)

13 BasicPropertiesofAlgebra(EqualityandCongruence,AdditionandMultiplication)14 Inductivevs.DeductiveReasoning

15 AnApproachtoProofs

Chapter3:ParallelandPerpendicularLines

16 ParallelLinesandTransversals

17 MultipleSetsofParallelLines

18 ProvingLinesareParallel

19 ParallelandPerpendicularLinesintheCoordinatePlane

Chapter4:Triangles Basic

20 T esofTrian les Scalene Isosceles E uilateral Ri ht21 CongruentTriangles(SAS,SSS,ASA,AAS,CPCTC)

22 CentersofTriangles

23 LengthofHeight,MedianandAngleBisector

24 InequalitiesinTriangles

Chapter5:Polygons

,

26 PolygonsMoreDefinitions(Definitions,DiagonalsofaPolygon)

27 InteriorandExteriorAnglesofaPolygon

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GeometryHandbooka eo ontents

Page Description

Chapter6:Quadrilaterals

29 FiguresofQuadrilaterals

30 CharacteristicsofParallelograms

31 ParallelogramProofs(SufficientConditions)

32 KitesandTrapezoids

Chapter7:Transformations

n ro uc on o rans orma on

35 Reflection

36 Rotation37 Rotationby90 aboutaPoint(x0,y0)

40 Translation

41 Compositions

Chapter8:Similarity

42 RatiosInvolvingUnits

43 SimilarPolygons

44 ScaleFactorofSimilarPolygons

45 DilationsofPolygons

46 MoreonDilation

, ,48 ProportionTablesforSimilarTriangles

49 ThreeSimilarTriangles

Chapter9:RightTriangles

50 PythagoreanTheorem

51 PythagoreanTriples

pec a r ang es r ang e, r ang e

53 TrigonometricFunctionsandSpecialAngles

54 TrigonometricFunctionValuesinQuadrantsII,III,andIV

55 GraphsofTrigonometricFunctions

56 Vectors57 OperatingwithVectors

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GeometryHandbooka eo ontents

Page Description

Chapter10:Circles

59 AnglesandCircles

Chapter11:PerimeterandArea

60 PerimeterandAreaofaTriangle

61 MoreontheAreaofaTriangle

62 PerimeterandAreaofQuadrilaterals

er me eran reao enera o ygons

64 CircleLengthsandAreas

65 AreaofCompositeFigures

Chapter12:SurfaceAreaandVolume

66 Polyhedra

67 AHoleinEulersTheorem

68 PlatonicSolids

69 Prisms

70 Cylinders

71 SurfaceAreabyDecomposition

72 Pyramids

73 Cones

74 S heres75 SimilarSolids

76 SummaryofPerimeterandAreaFormulas2DShapes

77 SummaryofSurfaceAreaandVolumeFormulas3DShapes

78 Index

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GeometryHandbooka eo ontentsUsefulWebsites

WolframMathWorldPerhapsthepremiersiteformathematicsontheWeb. Thissitecontains

definitions,explanationsandexamplesforelementaryandadvancedmathtopics.

http://mathworld.wolfram.com/

http://www.mathleague.com/help/geometry/geometry.htm

MathLeagueSpecializesinmathcontests,books,andcomputersoftwareforstudentsfromthe4th

gradethroughhighschool.

http://www.cde.ca.gov/ta/tg/sr/documents/rtqgeom.pdf

SchaumsOutlines

.

Agoodwaytotestyourknowledge.

Animportantstudentresourceforanyhighschoolmathstudentisa

SchaumsOutline. Eachbookinthisseriesprovidesexplanationsofthe

varioustopicsinthecourseandasubstantialnumberofproblemsforthe

studenttotry. Manyoftheproblemsareworkedoutinthebook,sothe

studentcanseeexamplesofhowtheyshouldbesolved.

Note: This study guide was prepared to be a companion to most books on the subject of High

SchaumsOutlinesareavailableatAmazon.com,Barnes&Noble,Bordersand

otherbooksellers.

. , , , ,

determine which subjects to include in this guide. Although a significant effort was made to make

the material in this study guide original, some material from Geometry was used in the preparation

of the study guide.

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Geometry

Points,Lines&Planes

Anintersectionofgeometric

shapesisthe

set

of

points

they

shareincommon.

landmintersectatpointE.

landnintersectatpointD.

mandnintersectinline .

Item Illustration Notation Definition

Point Alocationinspace.Segment Astraightpaththathastwoendpoints.Ray Astraightpaththathasoneendpointandextendsinfinitelyinonedirection.Line l or Astraightpaththatextendsinfinitelyin

bothdirections.

Plane m or Aflatsurfacethatextendsinfinitelyintwodimensions.Collinearpointsarepointsthatlieonthesameline.

Coplanarpointsarepointsthatlieonthesameplane.

Inthefi gure atright:

, , , , and arepoints.

lisaline

mandnareplanes.Inaddit n :io ,notethat

arecollinearpoints., , and arecoplanarpoints., d , d arecoplanarpoints.anan Ray goesoffinasoutheastdirection. Ray

goesof anorthwestdirection.fin

Together,rays and makeuplinel. Linelintersectsbothplanesmandn.

Note:Ingeometricfiguressuchastheoneabove,itis

importanttorememberthat,eventhoughplanesare

drawnwithedges,theyextendinfinitelyinthe2

dimensionsshown.

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Geometry

Segments,Rays&Lines

SomeThoughtsAbout

LineSegments

Linesegmentsaregenerallynamedbytheirendpoints,sothesegment ghtcouldbenamedeither or .atri

Segment containsthetwoendpoints(AandB)andallpointsonline thatarebetweenthem.

Rays

Raysaregenerallynamedbytheirsingleendpoint,calle ninitial

point,andanotherpointontheray.da

Ray containsitsinitialpointAandallpointsonline in edirectionofthearrow.th Rays and ar tthesameray.eno IfpointOisonline andisbetweenpointsAandB,

thenrays and arecalledoppositerays. TheyhaveonlypointOincommon,andtogethertheymakeupline .

Lines

Linesaregenerallynamedbyeitherasinglescriptletter(e.g., l)orbytwopointsontheline(e.g.,. ).

ow Alineextendsinfinitelyinthe directionssh nbyitsarrows.

Linesareparalleliftheyareinthesameplaneandtheyneverintersect. Linesfandg,atright,areparallel.

Linesareperpendiculariftheyintersectata90 angle. Apairofperpendicularlinesisalwaysinthesameplane.

Linesfande,atright,areperpendicular. Linesgande are

alsoperpendicular.

Linesareskewiftheyarenotinthesameplaneandtheyneverintersect. Lineskandl,atright,areskew.

(Rememberthisfigureis3dimensional.)

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Geometry

DistanceBetweenPoints

Distancemeasureshowfaraparttwothingsare. Thedistancebetweentwopointscanbe

measuredinanynumberofdimensions,andisdefinedasthelengthofthelineconnectingthe

twopoints. Distanceisalwaysapositivenumber.

1DimensionalDistance

Inonedimensionthedistancebetweentwopointsisdeterminedsimplybysubtractingthe

coordinatesofthepoints.

Example: Inthissegment,thedistancebetween 2and5iscalculatedas: 5 2 7.

2DimensionalDistance

Intwodimensions,thedistancebetweentwopointscanbecalculatedbyconsideringtheline

betweenthemtobethehypotenuseofarighttriangle. Todeterminethelengthofthisline:

Calculatethedifferenceinthexcoordinatesofthepoints Calculatethedifferenceintheycoordinatesofthepoints UsethePythagoreanTheorem.

Thisprocessisillustratedbelow,usingthevariabledfordistance.Example: Findthedistancebetween(1,1)and(2,5). Basedonthe

illustrationtotheleft:

x-coordinatedifference: 3.2 1y-coordinatedifference: 5 1 4.

Then,thedistanceiscalculatedusingtheformula:

4 9 16 25

3So, Ifwedefinetwopointsgenerallyas(x1,y1)and(x2,y2),thena2dimensionaldistanceformulawouldbe:

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ADVANCEDGeometryDistanceFormulainnDimensions

Thedistancebetweentwopointscanbegeneralizedtondimensionsbysuccessiveuseofthe

PythagoreanTheoreminmultipledimensions. Tomovefromtwodimensionstothree

dimensions,westartwiththetwodimensionalformulaandapplythePythagoreanTheoremto

addthethirddimension.

3Dimensions

Considertwo3dimensionalpoints(x1,y1,z1)and(x2,y2,z2). Considerfirstthesituationwherethetwozcoordinatesarethesame. Then,thedistancebetweenthepointsis2

dimensional,i.e., .Wethe thePythagoreanTheorem:naddathirddimensionusing

And,finallythe3dimensionaldifferenceformula:

nDimensions

Usingthesamemethodologyinndimensions,wegetthegeneralizedndimensional

difference e r n l, e sion):formula(wh retherea e termsbeneaththe radica oneforeachdim n

Or,inhigherlevelmathematicalnotation:

Thedistancebetween2pointsA=(a1,a2,,an)andB=(b1,b2,,bn)is

, | |

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Geometry

Angles

PartsofanAngle

Anangleconsistsoftworayswithacommon

endpoint(or,initialpoint).

Eachrayisasideoftheangle. Thecommonendpointiscalledthevertexof

theangle.

NamingAngles

Anglescanbenamedinoneoftwoways:

Pointvertexpointmethod. Inthismethod,theangleisnamedfromapointononeray,thevertex,andapointontheotherray. Thisisthemostunambiguousmethodof

naminganangle,andisusefulindiagramswithmultipleanglessharingthesamevertex.

Intheabovefigure,theangleshowncouldbenamedor .

Vertexmethod. Incaseswhereitisnotambiguous,ananglecanbenamedbasedsolelyonitsvertex. Intheabovefigure,theanglecouldbenamed.

MeasureofanAngle

Therearetwoconventionsformeasuringthesizeofanangle:

Indegrees. Thesymbolfordegreesis . Thereare360 inafullcircle. Theangleabovemeasuresapproximately45(oneeighthofacircle).

Inradians. Thereare2radiansinacompletecircle. Theangleabovemeasuresapproximately

radians.

SomeTermsRelatingtoAngles

Angleinterioristheareabetweentherays.

Angleexterior

isthe

area

not

between

the

rays.

Adjacentanglesareanglesthatsharearayforaside. and

inthefigureatrightareadjacentangles.

Congruentanglesareaangleswiththesamemeasure.

Anglebisectorisaraythatdividestheangleintotwocongruent

angles. Ray bisectsinthefigureatright.

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Geometry

TypesofAngles

SupplementaryAngles ComplementaryAngles

DC

A B

Angles mplementary.CandDareco

90AnglesAandBaresupplementary.

Angles linearpair.AandBforma

180

VerticalAngles

EF

G

H

Angleswhichareoppositeeachotherwhen

twolinescrossareverticalangles.

AnglesEandGareverticalangles.

Angles FandHareverticalangles.

In

addition,

each

angle

issupplementary

to

thetwoanglesadjacenttoit. Forexample:

AngleEissupplementarytoAnglesFandH.

Acute Obtuse

Right Straight

Anacuteangleisonethatislessthan90. In

theillustrationabove,anglesEandGare

acuteangles.

Arightangleisonethatisexactly90.

Anobtuseangleisonethatisgreaterthan

90. Intheillustrationabove,anglesFandH

areobtuseangles.

Astraightangleisonethatisexactly180.

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GeometryConditionalStatements

Aconditionalstatementcontainsbothahypothesisandaconclusioninthefollowingform:If

hypothesis,

then

conclusion.

Foranyconditionalstatement,itispossibletocreatethreerelatedconditionalstatements,asshownbelow. Inthetable,pisthehypothesisoftheoriginalstatementandqistheconclusionoftheoriginalstatement.

TypeofConditionalStatement ExampleStatementis:

OriginalStatement: Ifp,thenq. ( ) Example: Ifanumberisdivisibleby6,thenitisdivisibleby3. Theoriginalstatementmaybeeithertrueorfalse.

TRUE

ConverseStatement: Ifq,thenp. ( ) Example: Ifanumberisdivisibleby3,thenitisdivisibleby6. Theconversestatementmaybeeithertrueorfalse,andthisdoesnot

dependonwhethertheoriginalstatementistrueorfalse.FALSE

InverseStatement: Ifnotp,thennotq. (~ ~) Example: Ifanumberisnotdivisibleby6,thenitisnotdivisibleby3. Theinversestatementisalwaystruewhentheconverseistrueand

falsewhentheconverseisfalse.FALSE

ContrapositiveStatement: Ifnotq,thennotp. (~ ~) Example: Ifanumberisnotdivisibleby3,thenitisnotdivisibleby6. TheContrapositivestatementisalwaystruewhentheoriginal

statementistrueandfalsewhentheoriginalstatementisfalse.TRUE

Notealsothat: Whentwostatementsmustbeeitherbothtrueorbothfalse,theyarecalledequivalent

statements.o Theoriginalstatementandthecontrapositiveareequivalentstatements.o Theconverseandtheinverseareequivalentstatements.

Ifboththeoriginalstatementandtheconversearetrue,thephraseifandonlyif(abbreviatediff)maybeused. Forexample,Anumberisdivisibleby3iffthesumofitsdigitsisdivisibleby3.

Statementslinkedbelowbyredarrowsmustbeeitherbothtrueorbothfalse.

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GeometryBasicPropertiesofAlgebra

PropertiesofEqualityandCongruence.

PropertyDefinitionforEquality DefinitionforCongruence

Foranyrealnumbersa,b,andc: Foranygeometricelementsa,bandc.(e.g.,segment,angle,triangle)

ReflexiveProperty SymmetricProperty , , TransitiveProperty , , SubstitutionProperty If , then either can besubstituted for the other in any

equation (or inequality).

If , then either can be

substituted for the other in any

congruence expression.

MorePropertiesofEquality. Foranyrealnumbersa,b,andc:Property DefinitionforEquality

AdditionProperty , SubtractionProperty , MultiplicationProperty , DivisionProperty 0,

PropertiesofAdditionandMultiplication. Foranyrealnumbersa,b,andc:Property DefinitionforAddition DefinitionforMultiplication

CommutativeProperty AssociativeProperty DistributiveProperty

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GeometryInductivevs.DeductiveReasoning

InductiveReasoningInductivereasoningusesobservationtoformahypothesisorconjecture. Thehypothesiscanthenbetestedtoseeifitistrue. Thetestmustbeperformedinordertoconfirmthehypothesis.Example: Observethatthesumofthenumbers1to4is4 5/2andthatthesumofthenumbers1to5is5 6/2. Hypothesis:thesumofthefirstnnumbersis 1/2.Testingthishypothesisconfirmsthatitistrue.

DeductiveReasoningDeductivereasoningarguesthatifsomethingistrueaboutabroadcategoryofthings,itistrueofaniteminthecategory.

Example: Allbirdshavebeaks. Apigeonisabird;therefore,ithasabeak.Therearetwokeytypesofdeductivereasoningofwhichthestudentshouldbeaware:

LawofDetachment. Giventhat ,ifpistruethenqistrue. Inwords,ifonethingimpliesanother,thenwheneverthefirstthingistrue,thesecondmustalsobetrue.Example: Startwiththestatement:Ifalivingcreatureishuman,thenithasabrain.Thenbecauseyouarehuman,wecanconcludethatyouhaveabrain.

Syllogism. Giventhat and ,wecanconcludethat . Thisisakindoftransitivepropertyoflogic. Inwords,ifonethingimpliesasecondandthatsecondthingimpliesathird,thenthefirstthingimpliesthethird.Example: Startwiththestatements: Ifmypencilbreaks,Iwillnotbeabletowrite,andifIamnotabletowrite,Iwillnotpassmytest. ThenIcanconcludethatIfmypencilbreaks,Iwillnotpassmytest.

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GeometryAnApproachtoProofs

Learningtodevelopasuccessfulproofisoneofthekeyskillsstudentsdevelopingeometry.Theprocessisdifferentfromanythingstudentshaveencounteredinpreviousmathclasses,andmayseemdifficultatfirst. Diligenceandpracticeinsolvingproofswillhelpstudentsdevelopreasoningskillsthatwillservethemwellfortherestoftheirlives.RequirementsinPerformingProofs

Eachproofstartswithasetofgivens,statementsthatyouaresuppliedandfromwhichyoumustderiveaconclusion. Yourmissionistostartwiththegivensandtoproceedlogicallytotheconclusion,providingreasonsforeachstepalongtheway.

Eachstepinaproofbuildsonwhathasbeendevelopedbefore. Initially,youlookatwhatyoucanconcludefromthegivens. Thenasyouproceedthroughthestepsintheproof,youareabletouseadditionalthingsyouhaveconcludedbasedonearliersteps.

Eachstepinaproofmusthaveavalidreasonassociatedwithit. So,eachstatementintheproofmustbefurnishedwithananswertothequestion:Whyisthisstepvalid?

TipsforSuccessfulProofDevelopment Ateachstep,thinkaboutwhatyouknowandwhatyoucanconcludefromthat

information. Dothisinitiallywithoutregardtowhatyouarebeingaskedtoprove. Thenlookateachthingyoucanconcludeandseewhichonesmoveyouclosertowhatyouaretryingtoprove.

Goasfarasyoucanintotheprooffromthebeginning. Ifyougetstuck, Workbackwardsfromtheendoftheproof. Askyourselfwhatthelaststepintheproof

islikelytobe. Forexample,ifyouareaskedtoprovethattwotrianglesarecongruent,trytoseewhichoftheseveraltheoremsaboutthisismostlikelytobeusefulbasedonwhatyouweregivenandwhatyouhavebeenabletoprovesofar.

Continueworkingbackwardsuntilyouseestepsthatcanbeaddedtothefrontendoftheproof. Youmayfindyourselfalternatingbetweenthefrontendandthebackenduntilyoufinallybridgethegapbetweenthetwosectionsoftheproof.

Dontskipanysteps. Somethingsappearobvious,butactuallyhaveamathematicalreasonforbeingtrue. Forexample, mightseemobvious,butobviousisnotavalidreasoninageometryproof. Thereasonfor isapropertyofalgebracalledthereflexivepropertyofequality. Usemathematicalreasonsforallyoursteps.

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GeometryParallelLinesandTransversals

CorrespondingAnglesCorrespondingAnglesareanglesinthesamelocationrelativetotheparallellinesandthetransversal. Forexample,theanglesontopoftheparallellinesandleftofthetransversal(i.e.,topleft)arecorrespondingangles.AnglesAandE(topleft)areCorrespondingAngles. SoareanglepairsBandF(topright),CandG(bottomleft),andDandH(bottomright). Correspondinganglesarecongruent.AlternateInteriorAnglesAnglesDandEareAlternateInteriorAngles. AnglesCandFarealsoalternateinteriorangles.Alternateinterioranglesarecongruent.AlternateExteriorAnglesAnglesAandHareAlternateExteriorAngles. AnglesBandGarealsoalternateexteriorangles. Alternateexterioranglesarecongruent.ConsecutiveInteriorAnglesAnglesCandEareConsecutiveInteriorAngles. AnglesDandFarealsoconsecutiveinteriorangles. Consecutiveinterioranglesaresupplementary.

Notethatangles A,D,E,andHarecongruent,andanglesB,C,F,andGarecongruent. Inaddition,eachoftheanglesinthefirstgrouparesupplementarytoeachoftheanglesinthesecondgroup.

Transversal

HGFE

C DBA

Alternate:referstoanglesthatareonoppositesidesofthetransversal.Consecutive:referstoanglesthatareonthesamesideofthetransversal.Interior:referstoanglesthatarebetweentheparallellines.Exterior:referstoanglesthatareoutsidetheparallellines.

ParallelLines

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GeometryMultipleSetsofParallelLines

TwoTransversalsSometimes,

the

student

is

presented

two

sets

of

intersecting

parallel

lines,

as

shown

above.

Notethateachpairofparallellinesisasetoftransversalstotheothersetofparallellines.

GE F

H POM N

KI J

LDCBA

Inthiscase,thefollowinggroupsofanglesarecongruent:

Group1:AnglesA,D,E,H,I,L,MandPareallcongruent. Group2:AnglesB,C,F,G,J,K,N,andOareallcongruent. EachangleintheGroup1issupplementarytoeachangleinGroup2.

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GeometryProvingLinesareParallel

Thepropertiesofparallellinescutbyatransversalcanbeusedtoprovetwolinesareparallel.

CorrespondingAnglesIftwolinescutbyatransversalhavecongruentcorrespondingangles,

thenthelinesareparallel. Notethatthereare4setsofcorresponding

angles.

AlternateInteriorAnglesIftwolinescutbyatransversalhavecongruentalternateinteriorangles

congruent,then

the

lines

are

parallel.

Note

that

there

are

2sets

of

alternateinteriorangles.

AlternateExteriorAnglesIftwolinescutbyatransversalhavecongruentalternateexterior

angles,thenthelinesareparallel. Notethatthereare2setsof

alternateexteriorangles.

ConsecutiveInteriorAnglesIftwolinescutbyatransversalhavesupplementaryconsecutive

interiorangles,thenthelinesareparallel. Notethatthereare2setsof

consecutiveinteriorangles.

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GeometryParallelandPerpendicularLinesintheCoordinatePlane

ParallelLinesTwolines if theirslopesareequal.areparallel

In form,ifthevaluesofarethesame.

Example: 2 3 and 2 1

InStandardForm,ifthecoefficientsofandareproportiona tweentheequations.lbe

Example: 3 and 2 5

6 4 7 Also,ifthelinesarebothvertical(i.e.,their

slopesareundefin de ).

Example: and 3 2

PerpendicularLinesTwolinesareperpendiculariftheproductoftheirslopesis . Thatis,iftheslopeshavedifferentsignsand tiveinverses.aremultiplica

In form,thevaluesofmultiplytoget 1..

Example: and 6 5

3

InStandardForm,ifyouaddtheproductofthexcoefficientstotheproductofthey

coefficientsand

get

zero.

Example: and4 6 4 3 2 5 because 4 3 6 2 0

Also,ifonelineis isundefined)andonelineishorizontal(i.e., 0).vertical(i.e.,Example: and 6

3

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Geometry

TypesofTriangles

Scalene

Isosceles

Equilateral Right

60

60 60

AScaleneTrianglehas3sidesofdifferent

lengths. Becausethesidesareof

differentlengths,theanglesmustalsobe

ofdifferentmeasures.

ARight

Triangleis

one

that

contains

a90

angle. Itmaybescaleneorisosceles,but

cannotbeequilateral. Righttriangles

havesidesthatmeettherequirementsof

thePythagoreanTheorem.

AnEquilateral

Triangle

has

all

3sides

the

samelength(i.e.,congruent). Becauseall

3sidesarecongruent,all3anglesmust

alsobecongruent. Thisrequireseach

angletobe60.

AnIsoscelesTrianglehas2sidesthesame

length(i.e.,congruent). Becausetwo

sidesarecongruent,twoanglesmustalso

becongruent.

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Geometry

CongruentTriangles

Thefollowingtheoremspresentconditionsunderwhichtrianglesarecongruent.

SideAngleSide(SAS)Congruence

SAScongruencerequiresthecongruenceof

twosidesandtheanglebetweenthosesides.

NotethatthereisnosuchthingasSSA

congruence;thecongruentanglemustbe

betweenthetwocongruentsides.

SideSideSide(SSS)CongruenceSSScongruencerequiresthecongruenceofall

threesides. Ifallofthesidesarecongruent

thenalloftheanglesmustbecongruent. The

converseisnottrue;thereisnosuchthingas

AAAcongruence.

AngleSideAngle(ASA)Congruence

ASAcongruencerequiresthecongruenceof

twoanglesandthesidebetweenthoseangles.

Note:ASAandAAScombinetoprovidecongruenceoftwotriangleswheneveranytwoanglesandanyonesideofthetrianglesarecongruent.AngleAngleSide(AAS)Congruence

AAScongruencerequiresthecongruenceof

twoanglesandasidewhichisnotbetween

thoseangles.

CPCTC

CPCTCmeanscorrespondingpartsofcongruenttrianglesarecongruent. Itisavery

powerfultoolingeometryproofsandisoftenusedshortlyafterastepintheproofwhereapair

oftrianglesisprovedtobecongruent.

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Geometry

CentersofTriangles

Thefollowingareallpointswhichcanbeconsideredthecenterofatriangle.

Centroid(Medians)

Thecentroidistheintersectionofthethreemediansofatriangle. Amedianisa

linesegmentdrawnfromavertextothemidpointofthelineoppositethe

vertex.

Thecentroidislocated2/3ofthewayfromavertextotheoppositeside. Thatis,thedistancefromavertextothecentroidisdoublethelengthfromthecentroidtothemidpointoftheoppositeline.

Themediansofatrianglecreate6innertrianglesofequalarea.

Orthocenter(Altitudes)

Theorthocenteristheintersectionofthethreealtitudesofatriangle. An

altitudeisalinesegmentdrawnfromavertextoapointontheoppositeside

(extended,ifnecessary)thatisperpendiculartothatside.

Inanacutetriangle,theorthocenterisinsidethetriangle. Inarighttriangle,theorthocenteristherightanglevertex. Inanobtusetriangle,theorthocenterisoutsidethetriangle.

Circumcenter(PerpendicularBisectors)

Thecircumcenteristheintersectionofthe

perpendicularbisectorsofthethreesidesofthe

triangle. Aperpendicularbisectorisalinewhich

bothbisectsthesideandisperpendiculartothe

side. Thecircumcenterisalsothecenterofthe

circlecircumscribedaboutthetriangle.

EulerLine:Interestingly,

thecentroid,orthocenter

andcircumcenterofa

trianglearecollinear(i.e.,

lieonthesameline,

whichiscalledtheEuler

Line). Inanacutetriangle,thecircumcenterisinsidethetriangle. Inarighttriangle,thecircumcenteristhemidpointofthehypotenuse. Inanobtusetriangle,thecircumcenterisoutsidethetriangle.

Incenter(AngleBisectors)

Theincenteristheintersectionoftheanglebisectorsofthethreeanglesof

thetriangle. Ananglebisectorcutsanangleintotwocongruentangles,each

ofwhichishalfthemeasureoftheoriginalangle. Theincenterisalsothe

centerofthecircleinscribedinthetriangle.

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GeometryLengthofHeight,MedianandAngleBisector

HeightTheformulaforthelengthofaheightofatriangleisderivedfromHeronsformulafortheareaofatriangle:

where, ,and

,,arethelengthsofthesidesofthetriangle.

MedianTheformulaforthelengthofamedianofatriangleis:

where,,,arethelengthsofthesidesofthetriangle.

AngleBisectorTheformulaforthelengthofananglebisectorofatriangleis:

where,,,arethelengthsofthesidesofthetriangle.

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Geometry

InequalitiesinTriangles

Anglesandtheiroppositesidesintrianglesarerelated. Infact,thisisoftenreflectedinthe

labelingofanglesandsidesintriangleillustrations.

Anglesandtheiroppositesidesareoften

labeledwiththesameletter. Anuppercase

letterisusedfortheangleandalowercase

letterisusedfortheside.

Therelationshipbetweenanglesandtheiroppositesidestranslatesintothefollowingtriangle

inequalities:

If , then

If , then

Thatis,inanytriangle,

Thelargestsideisoppositethelargestangle. Themediumsideisoppositethemediumangle. Thesmallestsideisoppositethesmallestangle.

OtherInequalitiesinTriangles

TriangleInequality:

Thesumofthelengthsofanytwosidesofatriangle

isgreaterthanthelengthofthethirdside. Thisisacrucialelementin

decidingwh s r l .ether egmentsofany3lengthscanformat iang e

ExteriorAngleInequality: Themeasureofanexternalangleisgreaterthanthemeasureof

eitherofthetwononadj ow:acentinteriorangles. Thatis,inthefigurebel

Note:theExteriorAngleInequalityismuchlessrelevantthantheExteriorAngleEquality.

ExteriorAngleEquality: Themeasureofanexternalangleisequaltothesumofthemeasures

ofthetwonon interior hatis,inthefigurebelow:adjacent angles. T

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Geometry

Polygons Basics

BasicDefinitions

Polygon:aclosedpathofthreeormorelinesegments,where:

notwosideswithacommonendpointarecollinear,and eachsegmentisconnectedatitsendpointstoexactlytwoothersegments.

Side: asegmentthatisconnectedtoothersegments(whicharealsosides)toformapolygon.

Vertex: apointattheintersectionoftwosidesofthepolygon. (pluralform:vertices)

Diagonal: asegment,fromonevertextoanother,whichisnotaside.

Concave:Apolygoninwhichitispossibletodrawadiagonaloutsidethe

polygon. (Noticetheorangediagonaldrawnoutsidethepolygonat

right.) Concavepolygonsactuallylookliketheyhaveacaveinthem.

Convex: Apolygoninwhichitisnotpossibletodrawadiagonaloutsidethe

polygon. (Noticethatalloftheorangediagonalsareinsidethepolygon

atright.) Convexpolygonsappearmoreroundedanddonotcontain

caves.

NamesofSomeCommonPolygons

Number

ofSides

Name

of

Polygon

Number

ofSides

Name

of

Polygon

3 Triangle 9 Nonagon

4 Quadrilateral 10 Decagon

5 Pentagon 11 Undecagon

6 Hexagon 12 Dodecagon

7 Heptagon 20 Icosagon

8 Octagon n n-gon

Diagonal

Namesofpolygons

aregenerallyformed

fromtheGreek

language;however,

somehybridformsof

LatinandGreek(e.g.,

undecagon)have

creptintocommon

usage.

Vertex

Side

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Geometry

PolygonsMoreDefinitions

Definitions

Equilateral:apolygoninwhichallofthesidesareequalinlength.

Equiangular: apolygoninwhichalloftheangleshavethesame

measure.

Regular: apolygonwhichisbothequilateralandequiangular. That

is,aregularpolygonisoneinwhichallofthesideshavethesamelengthandalloftheangleshavethesamemeasure.

InteriorAngle: Anangleformedbytwosidesofapolygon. The

angleisinsidethepolygon.

ExteriorAngle: Anangleformedbyonesideofapolygonandthe

linecontaininganadjacentsideofthepolygon. Theangleisoutside

thepolygon.

Interior

AngleExterior

Angle

AdvancedDefinitions:

SimplePolygon: apolygonwhosesidesdonotintersectatanylocationotherthanitsendpoints. Simple

polygonsalwaysdividea

planeintotworegions

oneinsidethepolygonand

oneoutsidethepolygon.

ComplexPolygon: a

polygonwithsidesthatintersectsomeplaceotherthantheirendpoints(i.e.,notasimplepolygon).

Complexpolygonsdonot

alwayshavewelldefined

insidesandoutsides.

SkewPolygon: apolygonforwhichnotallofitsverticeslieonthesameplane.

HowManyDiagonalsDoesaConvexPolygonHave?

Believeitornot,thisisacommonquestionwithasimplesolution. Considerapolygonwithn

sidesand,therefore,nvertices.

Eachofthenverticesofthepolygoncanbeconnectedtootherverticeswithdiagonals.

That

is,

itcan

be

connected

to

all

other

vertices

except

itself

and

the

two

to

whichitisconnectedbysides. So,thereare linestobedrawnasdiagonals.

However,whenwedothis,wedraweachdiagonaltwicebecausewedrawitoncefromeachofitstwoendpoints. So,thenumberofdiagonalsisactuallyhalfofthenumberwe

calculatedabove.

Therefore,thenumberofdiagonalsin polygonis:annsided

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Geometry

InteriorandExteriorAnglesofaPolygon

InteriorAngles

Thesumofthe sidedpolygonis:interioranglesinan

InteriorAngles

Sides

Sumof

Interior

Angles

Each

Interior

Angle

3 180 60

4 360 90

5 540 108

6

720

120

7 900 129

8 1,080 135

9 1,260 140

10 1,440 144

Ifthepolygonisregular,youcancalculatethemeasureof

eachinteriorangleas:

ExteriorAngles

ExteriorAngles

Sides

Sumof

Exterior

Angles

Each

Exterior

Angle

3 360 120

4 360 90

5

360

72

6 360 60

7 360 51

8 360 45

9 360 40

10 360 36

Nomatterhowmanysidesthereareinapolygon,thesum

oftheexterioranglesis:

Ifthepolygonisregular,youcancalculatethemeasureof

eachexteriora s:nglea

Notation: TheGreekletterisequivalent

totheEnglishletterSandismathshorthand

forasummation(i.e.,addition)ofthings.

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GeometryDefinitionsofQuadrilaterals

Name DefinitionQuadrilateral Apolygonwith4sides.Kite Aquadrilateralwithtwoconsecutivepairsofcongruentsides,but

withoppositesidesnotcongruent.Trapezoid Aquadrilateralwithexactlyonepairofparallelsides.IsoscelesTrapezoid Atrapezoidwithcongruentlegs.Parallelogram

A

quadrilateral

with

both

pairs

of

opposite

sides

parallel.

Rectangle Aparallelogramwithallanglescongruent(i.e.,rightangles).Rhombus Aparallelogramwithallsidescongruent.Square Aquadrilateralwithallsidescongruentandallanglescongruent.

QuadrilateralTree:Quadrilateral

Kite Parallelogram Trapezoid

Rectangle Rhombus IsoscelesTrapezoid

Square

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GeometryFiguresofQuadrilaterals

IsoscelesTrapezoid 1pairofparallelsides Congruentlegs 2pairofcongruentbase

angles

Diagonalscongruent

Kite 2consecutivepairsof

congruentsides 1pairofcongruent

oppositeangles

Diagonalsperpendicular

Trapezoid

1pairofparallelsides(calledbases)

Anglesonthesamesideofthebasesaresupplementary

Parallelogram Bothpairsofoppositesidesparallel Bothpairsofoppositesidescongruent Bothpairsofoppositeanglescongruent Consecutiveanglessupplementary Diagonalsbisecteachother

Rectangle

Parallelogramwithallanglescongruent(i.e.,rightangles)

Diagonalscongruent

Rhombus Parallelogramwithallsidescongruent Diagonalsperpendicular Eachdiagonalbisectsapairof

oppositeangles

Square

BothaRhombusandaRectangle Allanglescongruent(i.e.,rightangles) Allsidescongruent

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GeometryCharacteristicsofParallelograms

Characteristic Square Rhombus Rec2pairofparallelsides

Oppositesidesarecongruent

Oppositeanglesarecongruent

Consecutiveanglesaresupplementary

Diagonals

bisect

each

other

All4anglesarecongruent(i.e.,rightangles)

Diagonalsarecongruent

All4sidesarecongruent

Diagonalsareperpendicular

Eachdiagonalbisectsapairofoppositeangles

Notes:Redmarksareconditionssufficienttoprovethequadrilateralisofthetypespecified.

Greenmarksareconditionssufficienttoprovethequadrilateralisofthetypespecifiedifthe

parallelogram.

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GeometryParallelogramProofs

Provinga

Quadrilateral

is

a

Parallelogram

Toproveaquadrilateralisaparallelogram,proveanyofthefollowingconditions:1. Bothpairsofoppositesidesareparallel. (note:thisisthedefinitionofaparallelogram)2. Bothpairsofoppositesidesarecongruent.3. Bothpairsofoppositeanglesarecongruent.4. Aninteriorangleissupplementarytobothofitsconsecutiveangles.5. Itsdiagonalsbisecteachother.6. Apairofoppositesidesisbothparallelandcongruent.

ProvingaQuadrilateralisaRectangleToproveaquadrilateralisarectangle,proveanyofthefollowingconditions:1. All4anglesarecongruent.2. Itisaparallelogramanditsdiagonalsarecongruent.

ProvingaQuadrilateralisaRhombusToproveaquadrilateralisarhombus,proveanyofthefollowingconditions:1. All4sidesarecongruent.2. ItisaparallelogramandItsdiagonalsareperpendicular.3. Itisaparallelogramandeachdiagonalbisectsapairofoppositeangles.

ProvingaQuadrilateralisaSquareToproveaquadrilateralisasquare,prove:1. ItisbothaRhombusandaRectangle.

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GeometryKitesandTrapezoids

FactsaboutaKiteToproveaquadrilateralisakite,prove: Ithastwopairofcongruentsides. Oppositesidesarenotcongruent.Also,ifaquadrilateralisakite,then: Itsdiagonalsareperpendicular Ithasexactlyonepairofcongruentoppositeangles.

PartsofaTrapezoidBase

LegLeg

MidsegmentTrapezoidABCDhasthefollowingparts: and arebases. andarelegs.

isth idsegment.em and arediagonals.

Base Diagonals AnglesAandDformapairofbaseangles. AnglesBandCformapairofbaseangles.

TrapezoidMidsegmentTheoremThemidsegmentofatrapezoidisparalleltoeachofitsbasesand:

.

Provinga

Quadrilateral

is

an

Isosceles

Trapezoid

Toproveaquadrilateralisanisoscelestrapezoid,proveanyofthefollowingconditions:1. Itisatrapezoidandhasapairofcongruentlegs.(definitionofisoscelestrapezoid)2. Itisatrapezoidandhasapairofcongruentbaseangles.3. Itisatrapezoidanditsdiagonalsarecongruent.

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Geometry

IntroductiontoTransformation

ATransformationisamappingofthepreimageofageometricfigureontoanimagethat

retainskeycharacteristicsofthepreimage.

Definitions

ThePreImageisthegeometricfigurebeforeithasbeentransformed.

TheImageisthegeometricfigureafterithasbeentransformed.

Amappingisanassociationbetweenobjects. Transformationsaretypesofmappings. Inthe

figuresbelow,wesayABCDismappedontoABCD,or . Theorderoftheverticesiscriticaltoaproperlynamedmapping.

AnIsometry

isaone

to

one

mapping

that

preserves

lengths.

Transformations

that

are

isometries(i.e.,preservelength)arecalledrigidtransformations.

IsometricTransformations

Rotationisturninga

figurearoundapoint.

Rotatedfiguresretain

theirsizeandshape,but

nottheirorientation.

Reflectionisflippinga

figureacrossalinecalled

amirror. Thefigure

retainsitssizeandshape,

butappearsbackwards

afterthe

reflection.

Translationisslidinga

figureintheplanesothat

itchangeslocationbut

retainsitsshape,sizeand

orientation.

TableofCharacteristicsofIsometricTransformations

Transformation Reflection Rotation Translation

Isometry(RetainsLengths)? Yes Yes Yes

RetainsAngles? Yes Yes Yes

RetainsOrientationtoAxes? No No Yes

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Geometry

IntroductiontoTransformation(contd)

TransformationofaPoint

Apoint

isthe

easiest

object

to

transform.

Simply

reflect,

rotate

or

translate

itfollowing

the

rulesforthetransformationselected. Bytransformingkeypointsfirst,anytransformation

becomesmucheasier.

TransformationofaGeometricFigure

Totransformanygeometricfigure,itisonlynecessarytotransformtheitemsthatdefinethe

figure,andthenreformit. Forexample:

Totransformalinesegment,transformitstwoendpoints,andthenconnecttheresultingimageswithalinesegment.

To

transform

aray,

transform

the

initial

point

and

any

other

point

on

the

ray,

and

then

constructarayusingtheresultingimages.

Totransformaline,transformanytwopointsontheline,andthenfitalinethroughtheresultingimages.

Totransformapolygon,transformeachofitsvertices,andthenconnecttheresultingimageswithlinesegments.

Totransformacircle,transformitscenterand,ifnecessary,itsradius. Fromtheresultingimages,constructtheimagecircle.

Totransformotherconicsections(parabolas,ellipsesandhyperbolas),transformthefoci,verticesand/ordirectrix. Fromtheresultingimages,constructtheimageconic

section.

Example: ReflectQuadrilateralABCD

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Geometry

Reflection

Definitions

Reflectionisflipping

afigure

across

amirror.

TheLineofReflectionisthemirrorthroughwhichthe

reflectiontakesplace.

Notethat:

Thelinesegmentconnectingcorrespondingpointsintheimageandpreimageisbisectedbythemirror.

Thelinesegmentconnectingcorrespondingpointsintheimageandpreimageisperpendiculartothemirror.

ReflectionthroughanAxisortheLine

Reflectionofthepoint(a,b)throughthex- ory-axisortheline givesthefollowingresults:

PreImage

Point

Mirror

Line

Image

Point

(a, b) xaxis (a, b)

(a, b) yaxis (a, b)

(a, b) the line: (a, b)

Ifyouforgettheabo le,startwith onasetof ateaxes. Reflectthepointthroughtheselectedlineandseewhichsetofa,bcoordinatesworks.

vetab thepoint 3,2 coordin

LineofSymmetry

ALineofSymmetryisanylinethroughwhichafigurecanbemappedontoitself. Thethinblack

linesinthefollowingfiguresshowtheiraxesofsymmetry:

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Geometry

Rotation

Definitions

Rotationisturning

afigure

by

an

angle

about

afixed

point.

TheCenterofRotationisthepointaboutwhichthefigureis

rotated. PointP,atright,isthecenterofrotation.

TheAngleofRotationdeterminestheextentoftherotation.

Theangleisformedbytheraysthatconnectthecenterof

rotationtothepreimageandtheimageoftherotation.Angle

P,atright,istheangleofrotation. Thoughshownonlyfor

PointA,theangleisthesameforanyofthefigures4vertices.

Note: Inperformingrotations,itisimportanttoindicatethedirectionoftherotation

clockwiseorcounterclockwise.

RotationabouttheOrigin

Rotationofthepoint(a,b)abouttheorigin(0,0)givesthefollowingresults:

PreImage

Point

Clockwise

Rotation

Counterclockwise

Rotation

Image

Point

(a, b) 90 270 (b, a)

(a, b) 180 180 (a, b)

(a, b) 270 90 (b, a)

(a, b) 360 360 (a, b)

Ifyouforg abovetable,star thepoint3,2 on tofcoordinatea otatethepointbytheselectedangleandseewhichsetofa,bcoordinatesworks.

etthe twith ase xes. R

RotationalSymmetry

AfigureinaplanehasRotationalSymmetryifitcanbemappedontoitselfbyarotationof

180 orless. Anyregularpolygonhasrotationalsymmetry,asdoesacircle. Herearesomeexamplesoffigureswithrotationalsymmetry:

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ADVANCED

Geometry

Rotationby90 aboutaPoint(x0,y0)

Rotatinganobjectby90 aboutapointinvolvesrotatingeachpointoftheobjectby90 about

thatpoint. Forapolygon,thisisaccomplishedbyrotatingeachvertexandthenconnecting

themtoeachother,soyoumainlyhavetoworryaboutthevertices,whicharepoints. The

mathematicsbehindtheprocessofrotatingapointby90 isdescribedbelow:

Letsdefinethefollowingpoints:

Thepointaboutwhichtherotationwilltakeplace:(x0,y0) Theinitialpoint(beforerotation):(x1,y1) Thefinalpoint(afterrotation):(x2,y2)

Theproblemistodetermine(x2,y2)ifwearegiven(x0,y0)and(x1,y1). Itinvolves3steps:

1. Converttheproblemtooneofrotatingapointabouttheorigin(amucheasierproblem).

2. Performtherotation.3. Converttheresultbacktotheoriginalsetofaxes.

Wellconsidereachstepseparatelyandprovideanexample:

Problem:Rotateapointby90 aboutanotherpoint.

Step1:Converttheproblemtooneofrotatingapointabouttheorigin:

First,weaskhowthepoint(x1,y1)relatestothepointaboutwhichitwillberotated(x0,

y0)andcreateanew(translated)point. Thisisessentiallyanaxistranslation,which

wewillreverseinStep3.

GeneralSituation Example

PointsintheProblem

RotationCenter:(x0,y0) Initialpoint:(x1,y1) Finalpoint:(x2,y2)

PointsintheProblem

RotationCenter:(2,3) Initialpoint:(2,1) Finalpoint:tobedetermined

Calculateanewpointthatrepresentshow

(x1,y1)relatesto(x0,y0). Thatpointis:

(x1x0,y1y0)

Calculateanewpointthatrepresentshow

(2,1)relatesto(2,3). Thatpointis:

(4, 2)

Thenextstepsdependonwhetherwearemakingaclockwiseorcounterclockwiserotation.

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ADVANCED

Geometry

Rotationby90 aboutaPoint(contd)

ClockwiseRotation:

Step2:Performtherotationabouttheorigin:

Rotatingby90 clockwiseabouttheorigin(0,0)issimplyaprocessofswitchingthex

andyvaluesofapointandnegatingthenewyterm. Thatis(x,y)becomes(y, x)after

rotationby90.


Prerotatedpoint(fromStep1):

(x1x0,y1y0)

Pointafterrotation:

(y1y0, x1+x0)


(4, 2)

Pointafterrotation:

(2,4)

Step3:Converttheresultbacktotheoriginalsetofaxes.

Todothis,simplyaddbackthepointofrotation(whichwassubtractedoutinStep1.


Pointafterrotation:

(y1y0, x1+x0)

Addbackthepointofrotation(x0,y0):

(y1y0+x0

, x1+x0+y0)

whichgivesusthevaluesof(x2,y2)

Pointafterrotation:

(2,4)

Addbackthepointofrotation(2,3):

(0,7)

Finally,lookattheformulasforx2andy2:

Clockwise Rotation

x2= y1 - y0+ x0

y2 = -x1 + x0+ y0

Noticethattheformulasfor

clockwiseandcounter

clockwiserotationby90 are

thesameexceptthetermsin

blueare

negated

between

the

formulas.

Interestingnote: Ifyouareaskedtofindthepointaboutwhichtherotationoccurred,you

simplysubstituteinthevaluesforthestartingpoint(x1,y1)andtheendingpoint(x2,y2)and

solvetheresultingpairofsimultaneousequationsforx0andy0.

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ADVANCED

Geometry

Rotationby90 aboutaPoint(contd)

CounterClockwise

Rotation:

Step2:Performtherotationabouttheorigin:

Rotatingby90 counterclockwiseabouttheorigin(0,0)issimplyaprocessofswitching

thex andyvaluesofapointandnegatingthenewxterm. Thatis(x,y)becomes(y,x)

afterrotationby90.



(x1x0,y1y0)

Pointafterrotation:(y1+y0,x1x0)


(4, 2)

Pointafterrotation:(2, 4)

Step3:Converttheresultbacktotheoriginalsetofaxes.

Todothis,simplyaddbackthepointofrotation(whichwassubtractedoutinStep1.


Pointafterrotation:

(y1+y0,x1x0)

Addbackthepointofrotation(x0,y0):

(y1+y0+x0,x1x0+y0)

whichgivesusthevaluesof(x2,y2)

Pointafterrotation:

(2,4)

Addbackthepointofrotation(2,3):

(4, 1)

Finally,lookattheformulasforx2andy2:

Noticethattheformulasfor

clockwiseandcounter

clockwiserotationby90 are

thesameexceptthetermsin

bluearenegatedbetweenthe

formulas.

Counter-Clockwise Rotation

x2= -y1 + y0+ x0

y2 = x1 - x0+ y0

Interestingnote: Thepointhalfwaybetweentheclockwiseandcounterclockwiserotationsof

90 isthecenterofrotationitself,(x0,y0). Intheexample,(2,3)ishalfwaybetween(0,7)and

(4,1).

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Geometry

Translation

Definitions

WhenTwoReflections OneTranslation

Translationissliding

afigure

in

the

plane.

Each

pointinthefigureismovedthesamedistancein

thesamedirection. Theresultisanimagethat

looksthesameasthepreimageineveryway,

exceptithasbeenmovedtoadifferentlocation

intheplane.

Eachofthefourorangelinesegmentsinthe

figureatrighthasthesamelengthanddirection.

Iftwomirrorsareparallel,thenreflectionthrough

oneofthem,followedbyareflectionthroughthe

secondisatranslation.

Inthefigureatright,theblacklinesshowthepaths

ofthetworeflections;thisisalsothepathofthe

resultingtranslation. Notethefollowing:

Thedistanceoftheresultingtranslation(e.g.,fromAtoA)isdoublethedistance

betweenthemirrors.

Theblacklinesofmovementareperpendiculartobothmirrors.

DefiningTranslationsintheCoordinatePlane(UsingVectors)

Atranslationmoveseachpointbythesamedistanceinthesamedirection. Inthecoordinate

plane,thisisequivalenttomovingeachpointthesameamountinthex-directionandthesame

amountinthey-direction. Thiscombinationofx-andy-directionmovementisdescribedbya

mathematicalconceptcalledavector.

Intheabovefigure,translationfromAtomoves10inthex-directionandthe-3inthey-direction. Invectornotation,thisis: ,. Noticethehalfraysymboloverthetwopointsandthefunnylookingbracketsaroundthemovementvalues.

So,thetranslationresultingfromthetworeflectionsintheabovefiguremoveseachpointof

thepreimagebythevector . Everytranslationcanbedefinedbythevectorrepresentingitsmovementinthecoordinateplane.

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Geometry

Compositions

Whenmultipletransformationsarecombined,theresultiscalledaCompositionofthe

Transformations. Twoexamplesofthisare:

Combiningtworeflectionsthroughparallelmirrorstogenerateatranslation(seethepreviouspage).

Combiningatranslationandareflectiontogeneratewhatiscalledaglidereflection.Theglidepartofthenamereferstotranslation,whichisakindofglidingofafigureon

theplane.

Note:Inaglidereflection,ifthelineofreflectionisparalleltothedirectionofthe

translation,itdoesnotmatterwhetherthereflectionorthetranslationisperformedfirst.

Figure2:ReflectionfollowedbyTranslation.Figure1:TranslationfollowedbyReflection.

CompositionTheorem

ThecompositionofmultipleisometriesisasIsometry. Putmoresimply,iftransformationsthat

preservelengtharecombined,thecompositionwillpreservelength. Thisisalsotrueof

compositionsoftransformationsthatpreserveanglemeasure.

OrderofComposition

Ordermattersinmostcompositionsthatinvolvemorethanoneclassoftransformation. Ifyou

applymultipletransformationsofthesamekind(e.g.,reflection,rotation,ortranslation),order

generallydoesnotmatter;however,applyingtransformationsinmorethanoneclassmay

producedifferentfinalimagesiftheorderisswitched.

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Example:

Note:theunitinchescancelout,sotheansweris,not .

3 12

14

Example:3

2 3

2 12 3 24

18

GeometryRatiosInvolvingUnits

RatiosInvolvingUnitsWhensimplifyingratioscontainingthesameunits:

Simplifythefraction. Notice that the units disappear. They behave

just like factors; if the units exist in thenumeratoranddenominator,thecancelandarenotintheanswer.

Whensimplifyingratioscontainingdifferentunits: Adjusttheratiosothatthenumeratoranddenominatorhavethesameunits. Simplifythefraction. Noticethattheunitsdisappear.

Dealingwith

Units

Noticeintheaboveexamplethatunitscanbetreatedthesameasfactors;theycanbeusedinfractions and they cancel when they divide. This fact can be used to figure out whethermultiplicationordivisionisneededinaproblem. Considerthefollowing:Example: Howlongdidittakeforacartravelingat48milesperhourtogo32miles?Considertheunitsofeachitem: 32 48

Ifyoumultiply,youget: 32 48 1,536

. Thisisclearlywrong! Ifyoudivide,youget: 32 48 . Now,

thislooksreasonable. Noticehowthe""unitcanceloutinthefinalanswer.Now you could have solved this problem by remembering that , or . However,payingcloseattention to theunitsalsogenerates thecorrectanswer. Inaddition,theunitstechniquealwaysworks,nomatterwhattheproblem!

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Geometry

SimilarPolygons

Insimilarpolygons,

Correspondinganglesarecongruent,and Correspondingsidesareproportional.

Bothoftheseconditionsarenecessaryfortwo

polygonstobesimilar. Conversely,whentwo

polygonsaresimilar,allofthecorresponding

anglesarecongruentandallofthesidesareproportional.

NamingSimilarPolygons

Similarpolygonsshouldbenamedsuchthatcorrespondinganglesareinthesamelocationin

thename,andtheorderofthepointsinthenameshouldfollowthepolygonaround.

Example: Thepolygonsabove llowingnames:couldbeshownsimilarwiththefo

~ Itwouldalsobeacceptabletoshowthesimilarityas:

~ Anynamesthatpreservetheorderofthepointsandkeepscorrespondinganglesin

correspondinglocationsinthenameswouldbeacceptable.

Proportions

Onecommonproblemrelatingtosimilarpolygonsistopresentthreesidelengths,wheretwo

ofthesidescorrespond,andtoaskforthe g the side gth.len thof correspondingtothethirdlen

Example: Intheabovesimilarpolygons,if 20, 12, 6, ?Thisproblemissolvablewithproportions. Todosoproperly,itisimportanttorelate

correspondingitems he

p portion:

in

t ro

20 126 1 0olygonisrepresentedonthe ofbothproportionsNoticethattheleftp top andthattheleft

mostsegmentsofthetwopolygonsareintheleftfraction.

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Geometry

ScaleFactorsofSimilarPolygons

Fromthesimilarpolygonsbelow, following he thsofthesides:the isknownaboutt leng

Thatis,theratiosofcorrespondingsidesinthe

twopolygonsarethesameandtheyequal

someconstant,calledthescalefactorofthetwopolygons. Thevalueof,then,isallyouneedtoknowtorelatecorrespondingsidesin

thetwopolygons.

FindingtheMissingLength

Anytimethestudentisaskedtofindthemissinglengthinsimilarpolygons:

Lookfortwocorrespondingsidesforwhichthevaluesareknown. Calculatethevalueof. Usethevalueoftosolveforthemissinglength.

isameasureoftherelativesizeofthetwopolygons. Usingthisknowledge,itispossibletoputintowordsaneasilyunderstandablerelationshipbetweenthepolygons.

LetPolygon1betheonewhosesidesareinthenumeratorsofthefractions. LetPolygon2betheonewhosesides einthedenominatorsofthefractions.ar Then,itcanbesaidthatPolygon1is timesthesizeofthePolygon2.

Example: In eabo polygons,if 20, 12, 6, ?th vesimilarSeeingthatandrelate,calculate:

126 2

Thensolveforbased thevalueof:on 20 2 1 0o everysideinthAlso,since 2,thelength f ebluepolygonisdoublethelengthofits

correspondingsideintheorangepolygon.

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Geometry

DilationofPolygons

Adilationisaspecialcaseoftransformationinvolvingsimilarpolygons. Itcanbethoughtofas

atransformationthatcreatesapolygonofthesameshapebutadifferentsizefromtheoriginal.

Keyelementsofadilationare:

ScaleFactorThescalefactorofsimilarpolygonsistheconstantwhichrepresentstherelativesizesofthepolygons.

Ce sthepointfromwhichthedilationtakesplace.nterThecenteriNotethat and 1inordertogenerateasecondpolygon. Then, 0

If thedilationiscalledanenlargement. 1, If

1,thedilationiscalledareduction.

DilationswithCenter(0,0)

Incoordinategeometry,dilationsareoftenperformedwiththecenterbeingtheorigin0,0.Inthatcase,toobtainthedilationofapolygon:

Multiplythecoordinatesofeachvertexbythescalefactor,and Connecttheverticesofthedilationwithlinesegments(i.e.,connectthedots).

Examples:

Inthe

following

examples:

Thegreenpolygonistheoriginal. Thebluepolygonisthedilation. Thedashedorangelinesshowthemovementawayfrom

(enlargement)ortoward(reduction)thecenter,whichis

theorigininall3examples.

N :oticethat,ineachexample

Thisfactcanbeusedtoconstructdilationswhencoordinateaxes

arenotavailable. Alternatively,thestudentcoulddrawasetof

coordinateaxesasanaidtoperformingthedilation.

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ADVANCED

Geometry

MoreonDilation

DilationsofNonPolygons

Anygeometricfigurecanbedilated. Inthedilationofthe

greencircleatright,noticethat:

Thedilationfactoris2. Theoriginalcirclehascenter ndradius7,3a 5. Thedilatedcirclehascenter14,6andradius 10.

So,thecenterandradiusarebothincreasedbyafactorof 2. Thisistrueofanyfigureinadilationwiththecenterattheorigin. Allofthekeyelementsthatdefinethefigureare

increased

by

the

scale

factor

.

DilationswithCenter,Inthefiguresbelow,thegreenquadrilateralsaredilatedtotheblueoneswithascalefactorof 2. Noticethefollowing:

Inthefiguretotheleft,thedilationhascenter0,0,whereasinthefiguretotheright,thedilationhascenter4,3. Thesizeoftheresultingfigureisthesameinbothcases

(because 2inbothfigures),butthelocationisdifferent.Graphically,theseriesoftransformationsthatisequivalenttoadilationfromapoint,otherthantheoriginisshownbelow. Compar lresulttothefigureabove(right).ethefina

Step1:Translatetheoriginalfigureby,toresetthecenterattheorigin. Step2: Performthedilation. Step3: Translatethedilatedfigureby

,. Thesestepsareillustratedbelow.

Step1 Step3Step2

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Geometry

SimilarTriangles

Thefollowingtheoremspresentconditionsunderwhichtrianglesaresimilar.

SideAngle

Side

(SAS)

Similarity

SASsimilarity requirestheproportionality of

twosidesandthecongruenceoftheangle

betweenthosesides. Notethatthereisnosuch

thingasSSAsimilarity;thecongruentanglemust

bebetweenthetwoproportionalsides.

SideSideSide(SSS)Similarity

SSSsimilarityrequirestheproportionalityofall

threesides. Ifallofthesidesareproportional,

thenall

of

the

angles

must

be

congruent.

AngleAngle(AA)Similarity

AAsimilarityrequiresthecongruenceoftwo

anglesandthesidebetweenthoseangles.

SimilarTriangleParts

Insimilartriangles,

Correspondingsidesareproportional. Correspondinganglesarecongruent.

Establishingthepropernamesforsimilartrianglesiscrucialtolineupcorrespondingvertices.

Inthepictureabove,wecansay:

or

~ or ~ or ~~ or ~ or ~Allofthesearecorrectbecausetheymatchcorrespondingpartsinthenaming. Eachofthese

similaritiesimpliesthe b t riangles:followingrelationships e weenpartsofthetwot

and d an

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Geometry

ProportionTablesforSimilarTriangles

SettingUpaTableofProportions

Itisoftenusefultosetupatabletoidentifytheproperproportions

inasimilarity. Considerthefiguretotheright. Thetablemightlook

somethinglikethis:

Triangle LeftSide RightSide BottomSide

Top AB BC CABottom

DE EF FD

Thepurposeofatablelikethisistoorganizetheinformationyouhaveaboutthesimilar

trianglessothatyoucanreadilydeveloptheproportionsyouneed.

DevelopingtheProportions

Todevelopproportionsfromthetable:

Extractthecolumnsneededfromthetable:AB BCDE EF Alsofromtheabove

table,

Eliminatethetablelines. Replacethehorizontallineswithdivisionlines. Putanequalsig hetworesultingfractions:nbetweent

Solvingfortheunknownlengthofaside:

Youcanextractanytwocolumnsyoulikefromthetable. Usually,youwillhaveinformationon

lengthsofthreeofthesidesandwillbeaskedtocalculateafourth.

Lookinthetableforthecolumnsthatcontainthe4sidesinquestion,andthensetupyour

proportion. Substituteknownvaluesintotheproportion,andsolvefortheremainingvariable.

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Geometry

ThreeSimilarTriangles

Acommonproblemingeometryistofindthemissingvalueinproportionsbasedonasetof

threesimilar

triangles,

two

of

which

are

inside

the

third.

The

diagram

often

looks

like

this:

c

PythagoreanRelationships

Insidetriangleontheleft: Insidetriangleontheright: Outside(large)triangle:

SimilarTriangleRelationships

Becauseallthreetrianglesaresimilar,wehavetherelationshipsinthetablebelow. These

relationshipsarenotobviousfromthepicture,butareveryusefulinsolvingproblemsbasedon

theabovediagram. Usingsimilaritiesbetweenthetriangles,2atatime,weget:

Fromthetwoinsidetriangles

Fromtheinsidetriangleon

theleftandtheoutside

triangle

Fromtheinsidetriangleon

therightandtheoutside

triangle

or or or

Theheightsquared

=theproductof:

thetwopartsofthebase

Theleftsidesquared

=theproductof:

thepartofthebasebelowit

andtheentirebase

Therightsidesquared

=theproductof:

thepartofthebasebelowit

andtheentirebase

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GeometryPythagoreanTheorem

where,

aandbare the lengthsof the legsofarighttriangle,and

cisthelengthofthehypotenuse.

Inarighttriangle,thePythagorean heoremsays:T

Right,Acute,orObtuseTriangle?Inadditiontoallowingthesolutionofrighttriangles,thePythagoreanFormulacanbeusedto

determinewhetheratriangleisarighttriangle,anacutetriangle,oranobtusetriangle.

Todeterminewhetheratriangleisobtuse,right,oracute:

Arrangeth esidesfromlowtohigh;callthema,b,andc,inincreasingorderelengthsofth Calculate: , , and . Compare: vs. Usetheillustrationsbelowtodeterminewhichtypeoftriangleyouhave.

ObtuseTriangle

RightTriangle

AcuteTriangle

5 8 . 92 5 6 4 8 1

Example:Trianglewithsides:5,8,9

7 9 . 124 9 8 1 1 4 4


6 8 . 103 6 6 4 1 0 0


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Geometry

PythagoreanTriples

PythagoreanTheorem:

Pythagoreantriples

are

sets

of

3positive

integers

that

meet

the

requirements

of

the

PythagoreanTheorem. Becausethesesetsofintegersprovideprettysolutionstogeometry

problems,theyareafavoriteofgeometrybooksandteachers. Knowingwhattriplesexistcan

helpthestudentquicklyidentifysolutionstoproblemsthatmightotherwisetakeconsiderable

timetosolve.

345TriangleFamily 72425TriangleFamily

9 16 25 49 576 625

51213TriangleFamily 81517TriangleFamily

25 144 169 64 225 289

Sample

Triples

51213

102426

153639

...

50120130

Sample

Triples

345

6810

91215

121620

304050

Sample

Triples

72425

144850

217275

...

70240250

Sample

Triples

81517

163034

244551

...

80150170

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Geometry

SpecialTriangles

Therelationshipamongthelengthsofthesidesofatriangleisdependentonthemeasuresof

theanglesinthetriangle. Forarighttriangle(i.e.,onethatcontainsa90 angle),twospecial

casesareofparticularinterest. Theseareshownbelow:

454590 Triangle

1

1

306090 Triangle

2

1

Inarighttriangle,weneedtoknowthelengthsoftwosidestodeterminethelengthofthe

third. Thepoweroftherelationshipsinthespecialtrianglesliesinthefactthatweneedonly

knowthelengthofonesideofthetriangletodeterminethelengthsoftheothertwosides.

ExampleSideLengths

Ina454590 triangle,thecongruenceoftwo

anglesguaranteesthecongruenceofthetwo

legsofthetriangle. Theproportionsofthethree

sidesare: . Thatis,thetwolegshave

thesamelengthandthehypotenuseistimes

aslong

as

either

leg.

Ina306090 triangle,theproportionsofthe

threesidesare: . Thatis,thelongleg

istimesaslongastheshortleg,andthe

hypotenuseis

timesas

long

as

the

short

leg.

454590 Triangle

306090 Triangle

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Geometry

TrigFunctionsandSpecialAngles

TrigonometricFunctions

SpecialAngles

TrigFunctionsofSpecialAngles

Radians

Degrees

0 0 02

042

10

4 0

6 30

12

12

32

1

333

4 45

22

22

1

3 60

32

12

12

3

1 3

2 90

42

102

0 undefined

SOHCAHTOA

si n

sin sin

cos

cos cos

tan

tan

tan

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GeometryTrigonometricFunctionValuesinQuadrantsII,III,andIV

InquadrantsotherthanQuadrantI,trigonometricvaluesforanglesarecalculatedinthe

followingmanner:

DrawtheangleontheCartesianPlane. Calculatethemeasureoftheanglefromthex

axisto.

Findthevalueofthetrigonometricfunctionoftheangleinthepreviousstep.

Assignaorsigntothetrigonometricvaluebasedonthefunctionusedandthe

quadrantisin.

Examples:inQuadrantIICalculate: 180 For 120,baseyourworkon180 120 60

sin60

,so:

inQuadrantIIICalculate: 180For 210,baseyourworkon210 180 30

cos 30

,so:

inQuadrantIVCalculate: 360 For 315,baseyourworkon360 315 45

tan 45 1,so:

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Thesineandcosecantfunctionsareinverses. So:sin

1

csc and csc

1

sin

Thecosineandsecantfunctionsareinverses. So:cos

1

sec and sec

1

cos

Thetangentandcotangentfunctionsareinverses. So:tan

1

cot and cot

1

tan

GeometryGraphsofTrigonometricFunctions

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GeometryVectors

Definitions

Avector

isageometric

object

that

has

both

magnitude(length)anddirection.

TheTailofthevectoristheendoppositethearrow.Itrepresentswherethevectorismovingfrom.

TheHeadofthevectoristheendwiththearrow. Itrepresentswherethevectorismovingto.

TheZeroVectorisdenoted0. Ithaszerolengthandallthepropertiesofzero.

Twovectorsareequalistheyhaveboththesamemagnitudeandthesamedirection. Twovectorsareparalleliftheyhavethesameoroppositedirections. Thatis,iftheangles

ofthevectorsarethesameor180 different.

Twovectorsareperpendicularifthedifferenceoftheanglesofthevectorsis90 or270.MagnitudeofaVectorThedistanceformulagivesthemagnitudeofavector. Iftheheadandtailofvectorvarethe

points , and a e is, ,thenthem gnitud ofv :

|| Notethat . Thedirectionsofthetwovectorsareopposite,buttheirmagnitudesarethesame.

DirectionofaVectorThedirectionofavectorisdeterminedbytheangleitmakes

withahorizontalline. Inthefigureatright,thedirectionisthe

angle. Thevalueofcanbecalculatedbasedonthelengthsofthesidesofthetriang thevectorforms.le

or

wherethefunctiontan-1istheinversetangentfunction. Thesecondequationinthelineabove

readsistheanglewhosetangentis.

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GeometryOperationswithVectors

Itispossibletooperatewithvectorsinsomeofthesamewaysweoperatewithnumbers. In

particular:

AddingVectorsVectorscanbe in tangularformbyseparatelyaddingtheirx-andy-components. In

general,

added rec

, , , , ,

Example:Inthe figureatright,

4, 36 2,

4, 3 2,6 6,3

V aectorAlgebr a a a

a b a b 1 ab a b b a

ScalarMultiplicationScalarmultiplic sthemagnitudeofavector,butnotthedirection. Ingeneral,ationchange

, ,

Inthefigureatright,

4, 32 2 4, 3 8, 6

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GeometryPartsofCircles

Centerthemiddleofthecircle. Allpointsonthecirclearethesamedistancefromthecenter.

Radiusalinesegmentwithoneendpointatthecenter

andtheotherendpointonthecircle. Thetermradiusis

alsousedtorefertothedistancefromthecentertothe

pointsonthecircle.

Diameteralinesegmentwithendpointsonthecirclethatpassesthroughthecenter.

Arcapathalongacircle.MinorArcapathalongthecirclethatislessthan180.MajorArcapathalongthecirclethatisgreaterthan180.

Semicircleapathalongacirclethatequals180.Sectoraregioninsideacirclethatisboundedbytworadiiandanarc.

SecantLinealinethatintersectsthecircleinexactlyonepoint.

TangentLinealinethatintersectsthecircleinexactlytwopoints.

Chordalinesegmentwithendpointsonthecirclethatdoesnotpassthroughthecenter.

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GeometryAnglesandCircles

CentralAngle

Inscribed

Angle

Vertexinsidethecircle Vertexoutsidethecircle

Tangentononeside Tangentsontwosides

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GeometryPerimeterandAreaofaTriangle

PerimeterofaTriangleTheperimeterofatriangleissimplythesumofthemeasuresofthethreesidesofthetriangle.

AreaofaTriangleTherearetwoformulasfortheareaofatriangle,dependingonwhatinformationaboutthe

triangle

is

available.

Formula1: Theformulamostfamiliartothestudentcanbeusedwhenthebaseandheightofthetriangleareeitherknownorcanbedetermined.

where, isthelengthofthebaseofthetriangle.istheheightofthetriangle.

Note: Thebasecanbeanysideofthetriangle. Theheightisthemeasureofthealtitudeof

whicheverside

is

selected

as

the

base.

So,

you

can

use:

or or

Formula2: Heronsformulafortheareaofatrianglecanbeusedwhenthelengthsofallofthesidesareknown. Sometimesthisformula,

thoughlessappealing,canbeveryuseful.

where,

. Note: issometimescalledthesemiperimeterofthetriangle.

,,arethelengthsofthesidesofthetriangle.

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ADVANCEDGeometry

MoreontheAreaofaTriangle

TrigonometricFormulasThefollowingformulasfortheareaofatrianglecomefromtrigonometry.Whichoneisuseddependsontheinformationavailable:Twoanglesandaside:

Twosidesandanangle:

CoordinateGeometryIfthethreeverticesofatrianglearedisplayedinacoordinateplane,theformulabelow,usingadeterminant,willgivetheareaofatriangle.Letthethreepointsinthecoordinateplanebe:, , , , , . Then,theareaofthetriangleisonehalfoftheabsolutevalueofthedeterminantbelow:

Example:For

the

triangle

in

the

figure

at

right,

the

area

is:

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GeometryPerimeterandAreaofQuadrilaterals

Name Illustration PerimeterKite 2 2

Trapezoid

2 2Parallelogram

Rectangle 2 2

Rhombus 4

4Square

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GeometryPerimeterandAreaofRegularPolygons

DefinitionsRegularPolygons Thecenterofapolygonisthecenterofitscircumscribed

circle. PointOisthecenterofthehexagonatright. Theradiusofthepolygonistheradiusofits

circumscribedcircle. and arebothradiiofthehexagonatright.

Theapothemofapolygonisthedistancefromthecentertothemidpointofanyofitssides. aistheapothemofthehexagonatright.

Thecentralangleofapolygonisananglewhosevertexisthecenterofthecircleandwhosesidespassthroughconsecutiveverticesofthepolygon. Inthefigureabove,isacentralangleofthehexagon.

AreaofaReg P onular olyg where, istheapothemofthepolygon

istheperimeterofthepolygon

PerimeterandAreaofSimilarFiguresLetkbethescalefactorrelatingtwosimilargeometricfiguresF1andF2suchthat .

Then,

and

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GeometryCircleLengthsandAreas

Circum Areaferenceand

istheareaofthecircle.isthecircumference(i.e.,theperimeter)ofthecircle.

where: istheradiusofthecircle.

LengthofanArconaCircleAcommonprobleminthegeometryofcirclesistomeasurethelengthofanarconacircle.Definitio thecircumferenceofacircle.n:Anarcisasegmentalong

where: AB isthemeasure(indegrees)ofthearc. Notethat

thisisalsothemeasureofthecentralangle.

isthecircumferenceofthecircle.

AreaofaSectorofaCircleAnothercommonprobleminthegeometryofcirclesistomeasuretheareaofasectoracircle.Definitio a lethatisboundedbytworadiiandanarcofthecircle.n:Asectorisaregion in circ

where: AB isthemeasure(indegrees)ofthearc. Notethat

thisisalsothemeasureofthecentralangle.istheareaofthecircle.

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GeometryAreaofCompositeFigures

Tocalculatetheareaofafigurethatisacompositeofshapes,considereachshapeseparately.Example1:Calculatetheareaoftheblueregioninthefiguretotheright.Tosolvethis:

Recognizethatthefigureisthecompositeofarectangleandtwotriangles.

Disassemblethecompositefigureintoitscomponents. Calculatetheareaofthecomponents. Subtracttogettheareaofthecompositefigure.

Example2:Calculatetheareaoftheblueregioninthefiguretotheright.Tosolvethis:

Recognizethatthefigureisthecompositeofasquareandacircle.

Disassemblethecompositefigureintoitscomponents. Calculatetheareaofthecomponents. Subtracttogettheareaofthecompositefigure.

~ .

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GeometryPolyhedra

DefinitionsFaces

APolyhedronisa3dimensionalsolidboundedbyaseriesofpolygons.

Facesarethepolygonsthatboundthepolyhedron. AnEdgeisthelinesegmentattheintersectionoftwofaces. AVertexisapointattheintersectionoftwoedges. ARegularpolyhedronisoneinwhichallofthefacesarethe

sameregularpolygon.Vertices

AConvexPolyhedronisoneinwhichalldiagonalsarecontainedwithintheinteriorofthepolyhedron.

A

Concave

polyhedron

is

one

that

is

not

convex.

ACrossSectionistheintersectionofaplanewiththepolyhedron.Eulers oremTheLet: numberoffacesofapolyhedron. the

henumberofverticesofapolyhedron. t

thenumberofedgesofapolyhedron.

Then,for

any

polyhedron

that

does

not

intersect

itself,

CalculatingtheNumberofEdges

Edges

EulersTheoremExample:Thecubeabovehas

6faces 8vertices 12edges

Thenumberofedgesofapolyhedronisonehalfthenumberofsidesinthepolygonsit

comprises. Eachsidethatiscountedinthiswayissharedbytwopolygons;simplyaddingall

thesides

of

the

polygons,

therefore,

double

counts

the

number

of

edges

on

the

polyhedron.

Example: Considerasoccerball. Itispolyhedronmadeupof20

hexagonsand12pentagons. Thenthenumberofedgesis:

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Example:

Thecubewithatunnelinithas 16 32 16

so,

ADVANCEDGeometry

AHoleinEulersTheoremTopologyisabranchofmathematicsthatstudiesthepropertiesofobjectsthatarepreservedthroughmanipulationthatdoesnotincludetearing. Anobjectmaybestretched,twistedandotherwisedeformed,butnottorn. Inthisbranchofmathematics,adonutisequivalenttoacoffeecupbecausebothhaveonehole;youcandeformeitherthecuporthedonutandcreatetheother,likeyouareplayingwithclay.Alloftheusualpolyhedrahavenoholesinthem,soEulersEquationholds. Whathappensifweallowthepolyhedratohaveholesinthem? Thatis,whatifweconsidertopologicalshapesdifferentfromtheoneswenormallyconsider?

EulersCharacteristicWhenEulersEquationisrewrittenas ,thelefthandsideoftheequationiscalledtheEulerCharacteristic.

GeneralizedEulers

Theorem

Let: thenumberoffacesofapolyhedron.

thenumberofverticesofapolyhedron. thenumberofedgesofapolyhedron. thenumberofholesinthepolyhedron. is

calledthegenusoftheshape.Then,foranypolyhedronthatdoesnotintersectitself,

NotethatthevalueofEulersCharacteristiccanbenegativeiftheshapehasmorethanoneholeinit(i.e.,if 2)!

TheEulerCharacteristicofashapeis:

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GeometryPlatonicSolids

A PlatonicSolidisaconvexregularpolyhedronwithfacescomposedofcongruentconvexregularpolygons. Therefiveofthem:

KeyPropertiesofPlatonicSolidsItisinterestingtolookatthekeypropertiesoftheseregularpolyhedra.

Name Faces Vertices Edges TypeofFaceTetrahedron 4 4 6 TriangleCube 6 8 12 SquareOctahedron 8 6 12 TriangleDodecahedron 12 20 30 PentagonIcosahedron 20 12 30 Triangle

Noticethefollowingpatternsinthetable:

Allofthenumbersoffacesareeven. Onlythecubehasanumberoffacesthatisnotamultipleof4.

Allofthenumbersofverticesareeven. Onlytheoctahedronhasanumberoffacesthatisnotamultipleof4.

Thenumberoffacesandverticesseemtoalternate(e.g.,cube68vs.octahedron86). Allofthenumbersofedgesaremultiplesof6. Thereareonlythreepossibilitiesforthenumbersofedges6,12and30. Thefacesareoneof:regulartriangles,squaresorregularpentagons.

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GeometryPrisms

Definitions

APrismisapolyhedronwithtwocongruentpolygonalfacesthatlieinparallelplanes.

TheBasesaretheparallelpolygonalfaces. TheLateralFacesarethefacesthatarenotbases. TheLateralEdgesaretheedgesbetweenthelateralfaces. TheSlantHeightisthelengthofalateraledge. Notethat

alllateraledgesarethesamelength.

TheHeightistheperpendicularlengthbetweenthebases. ARightPrismisoneinwhichtheanglesbetweenthebasesandthe

lateraledgesarerightangles. Notethatinarightprism,theheightand

theslantheightarethesame.

AnObliquePrismisonethatisnotarightprism.RightHexagonal

Prism TheSurfaceAreaofaprismisthesumoftheareasofallitsfaces. TheLateralAreaofaprismisthesumoftheareasofitslateralfaces.

SurfaceAreaandVolume htPrismwhere,

of aRigSurfaceArea: LateralSA: Volume:

CavalierisPrincipleIftwosolidshavethesameheightandthesamecrosssectionalareaateverylevel,thenthey

havethesamevolume. Thisprincipleallowsustoderiveaformulaforthevolumeofan

obliqueprism

from

the

formula

for

the

volume

of

aright

prism.

SurfaceAreaandVolume bliquePrismwhere,

ofanOSurfaceArea: LateralSA: Volume:

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GeometryCylinders

Definitions

ACylinderisafigurewithtwocongruentcircularbasesinparallelplanes. AcylinderhasonlyoneLateralSurface. Whendeconstructed,thelateralsurfaceofa

cylinderisarectanglewithlengthequaltothecircumferenceofthebase.

TherearenoLateralEdgesinacylinder. TheSlantHeightisthelengthofthelateralsidebetweenthebases. Note

thatalllateraldistancesarethesamelength. Theslantheighthas

applicabilityonlyifthecylinderisoblique.

TheHeightistheperpendicularlengthbetweenthebases. ARightCylinderisoneinwhichtheanglesbetweenthebasesandthelateralsideareright

angles. Notethatinarightcylinder,theheightandtheslantheightarethesame.

AnObliqueCylinderisonethatisnotarightcylinder. TheSurfaceAreaofacylinderisthesumoftheareasofitsbasesanditslateralsurface. TheLateralAreaofacylinderistheareasofitslateralsurface.

SurfaceAreaandVolume htCylinderof aRigSurfaceArea:

where,

LateralSA: Volume:

SurfaceAreaandVolume bliquePrismofanOSurface

Area:

where,

LateralSA: Volume:

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GeometrySurfaceAreabyDecomposition

Sometimesthestudentisaskedtocalculatethesurfaceareofaprismthatdoesnotquitefitintooneofthecategoriesforwhichaneasyformulaexists. Inthiscase,theanswermaybetodecomposetheprismintoitscomponentshapes,andthencalculatetheareasofthecomponents. Note: thisprocessalsoworkswithcylindersandpyramids.DecompositionofaPrismTocalculatethesurfaceareaofaprism,decomposeitandlookateachoftheprismsfacesindividually.Example: Calculatethesurfaceareaofthetriangularprismatright.Todothis,firstnoticethatweneedthevalueofthehypotenuseofthebase. UsethePythagoreanTheoremorPythagoreanTriplestodeterminethemissingvalueis10. Then,decomposethefigureintoitsvariousfaces:

Thesurfacearea,then,iscalculatedas: 2

2 12 6 8 1 0 7 8 7 6 7 216DecompositionofaCylinder

Thesurfacearea,then,iscalculatedas: 2

2 3 6 5 48 ~ 150.80

Thecylinderatrightisdecomposedintotwocircles(thebases)andarectangle(thelateralface).

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GeometryPyramids

Pyramids APyramidisapolyhedroninwhichthebaseisapolygonand

thelateralsidesaretriangleswithacommonvertex.

TheBaseisapolygonofanysizeorshape. TheLateralFacesarethefacesthatarenotthebase. TheLateralEdgesaretheedgesbetweenthelateralfaces. TheApexofthepyramidistheintersectionofthelateral

edges. Itisthepointatthetopofthepyramid.

TheSlantHeightofaregularpyramidisthealtitudeofoneofthe

lateral

faces.

TheHeightistheperpendicularlengthbetweenthebaseandtheapex. ARegularPyramidisoneinwhichthelateralfacesarecongruenttriangles. Theheightofa

regularpyramidintersectsthebaseatitscenter.

AnObliquePyramidisonethatisnotarightpyramid. Thatis,theapexisnotaligneddirectlyabovethecenterofthebase.

TheSurfaceAreaofapyramidisthesumoftheareasofallitsfaces.

TheLateralAreaofapyramidisthesumoftheareasofitslateralfaces.

SurfaceAreaandVolume gularPyramidofaReSurfaceArea:

LateralSA:

Volume:

SurfaceAreaandVolume bliquePyramidofan O

SurfaceArea: Volume:

where,

where,

Thelateralsurfaceareaofanobliquepyramidisthesumof

theareasofthefaces,whichmustbecalculatedindividually.

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GeometryCones

Definitions

ACircularConeisa3dimensionalgeometricfigurewithacircularbasewhichtaperssmoothlytoavertex(orapex). Theapexandbaseareindifferentplanes. Note:thereis

alsoanellipticalconethathasanellipseasabase,butthatwillnotbeconsideredhere.

TheBaseisacircle. TheLateralSurfaceisareaofthefigurebetweenthebaseandtheapex. TherearenoLateralEdgesinacone. TheApexoftheconeisthepointatthetopofthecone. TheSlantHeightofaconeisthelengthalongthelateralsurfacefromtheapextothebase. TheHeightistheperpendicularlengthbetweenthebaseandtheapex. ARightConeisoneinwhichtheheightoftheconeintersectsthebaseat

itscenter.

AnObliqueConeisonethatisnotarightcone. Thatis,theapexisnotaligneddirectlyabovethecenterofthebase.

TheSurfaceAreaofaconeisthesumoftheareaofitslateralsurfaceanditsbase.

TheLateralAreaofaconeistheareaofitslateralsurface.SurfaceAreaandVolume tConeofaRigh

SurfaceArea: LateralSA: Volume:

SurfaceAreaandVolume iqueConeofan OblSurfaceArea: Volume:

where,

where,

Thereisnoeasyformulaforthelateralsurfaceareaofan

obliquecone.

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GeometrySpheres

Definitions

ASphereisa3dimensionalgeometricfigureinwhichallpointsareafixeddistancefromapoint. Agoodexampleof

asphereisaball.

Centerthemiddleofthesphere. Allpointsonthespherearethesamedistancefromthecenter.

Radiusalinesegmentwithoneendpointatthecenterandtheotherendpointonthesphere. Thetermradiusisalso

usedtorefertothedistancefromthecentertothepoints

onthe

sphere.

Diameteralinesegmentwithendpointsonthespherethatpassesthroughthecenter.

GreatCircletheintersectionofaplaneandaspherethatpassesthroughthecenter.

Hemispherehalfofasphere. Agreatcircleseparatesaplaneintotwohemispheres.

SecantLinealinethatintersectsthesphereinexactlyonepoint.

TangentLinealinethatintersectsthesphereinexactlytwopoints.

Chordalinesegmentwithendpointsonthespherethatdoesnotpassthroughthecenter.

SurfaceAreaandVolumeo fa SphereSurface

Area:

Volume:

where,

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GeometrySimilarSolids

SimilarSolidshaveequalratiosofcorrespondinglinearmeasurements(e.g.,edges,radii). So,alloftheirkeydimensionsareproportional.

Edges,SurfaceAreaandVolumeofSimilarFiguresLetkbethescalefactorrelatingtwosimilargeometricsolidsF1andF2suchthat .Then,forcorrespondingpartsofF1andF2,

and

And

Theseformulasholdtrueforanycorrespondingportionofthefigures. So,forexample:

T E L F

T E L F k A F F

A F F k

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GeometrySummaryofPerimeterandAreaFormulas2DShapes

Shape Figure Perimeter Area

Kite , ,

Trapezoid , ,

b, b basesh height

Parallelogram

,

Rectangle ,

Rhombus

,

Square

,

RegularPolygon

Circle

Ellipse

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Page Subject

16 AlternateExteriorAngles

16 AlternateInteriorAngles

23 AngleBisectorLengthinaTriangle

Angles

10 Angles Basic

11 Angles Types

Area

65 Area CompositeFigures

63 Area Polygons

62 Area Quadrilaterals

64 Area RegionofaCircle

60,61 Area Triangle

76 AreaFormulas Summaryfor2DShapes

69 Cavalieri'sPrinciple

CentersofTriangles

22 Centroid

22 Circumcenter

22 Incenter

22 Orthocenter

22 Centroid

Circles64 Circles ArcLengths

58 Circles DefinitionsofParts

64 Circles RegionAreas

59 Circles RelatedAngles

59 Circles RelatedSegments

22 CirclesandTriangles

22 Circumcenter

12 ConditionalStatements(Original,Converse,Inverse,Contrapositive)

Cones

73 Cones Definitions73 Cones Surfac

handbook of geometry

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