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8/13/2019 Handbook of Geometry
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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
October 30, 2013
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2/82
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.
GeneralSituation Example
Prerotatedpoint(fromStep1):
(x1x0,y1y0)
Pointafterrotation:
(y1y0, x1+x0)
Prerotatedpoint(fromStep1):
(4, 2)
Pointafterrotation:
(2,4)
Step3:Converttheresultbacktotheoriginalsetofaxes.
Todothis,simplyaddbackthepointofrotation(whichwassubtractedoutinStep1.
GeneralSituation Example
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.
GeneralSituation Example
Prerotatedpoint(fromStep1):
(x1x0,y1y0)
Pointafterrotation:(y1+y0,x1x0)
Prerotatedpoint(fromStep1):
(4, 2)
Pointafterrotation:(2, 4)
Step3:Converttheresultbacktotheoriginalsetofaxes.
Todothis,simplyaddbackthepointofrotation(whichwassubtractedoutinStep1.
GeneralSituation Example
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
Example:Trianglewithsides:7,9,12
6 8 . 103 6 6 4 1 0 0
Example:Trianglewithsides:6,8,10
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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