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Chapter10‐Similarity
Def:Theratioofthenumberatothenumberbisthenumber .
Aproportionisanequalitybetweenratios.
a,b,c,anddarecalledthefirst,second,third,andfourthterms. Thesecondandthirdterms,bandc,arecalledthemeans. Thefirstandfourthterms,aandd,arecalledtheextremes.
Theproductofthemeansisequaltotheproductoftheextremes.
If ,then .
Def:Thenumberbisthegeometricmeanbetweenthenumbersaandcifa,b,andcarepositiveand .
Def:Twotrianglesaresimilariffthereisacorrespondencebetweentheirverticessuchthattheircorrespondingsidesareproportionalandtheircorrespondinganglesareequal.
Theorem44‐TheSide‐SplitterTheoremIfalineparalleltoonesideofatriangleintersectstheothertwosidesindifferentpoints,
itdividesthesidesinthesameratio,thatis,ifintriangleABC,DE||BC,then .
CorollarytotheSide‐SplitterTheorem:Ifalineparalleltoonesideofatriangleintersectstheothertwosidesindifferentpoints,itcutsoffsegmentsproportionaltothesides,that
is, and
Theorem45‐TheAATheorem‐Iftwoanglesofonetriangleareequaltotwoanglesofanothertriangle,thetrianglesaresimilar.
CorollarytotheAATheorem‐Twotrianglessimilartoathirdtrianglearesimilartoeachother.
Theorem46‐Correspondingaltitudesofsimilartriangleshavethesameratioasthatofthecorrespondingsides.
Given:∆ABC~∆DEF;BGandEHarealtitudes Prove:
Theorem47–Theratiooftheperimetersoftwosimilarpolygonsisequaltotheratioofthecorrespondingsides.
Given:∆ABC~∆A’B’C’ Prove:∆
∆ ’ ’ ’,where
Theorem48–Theratiooftheareasoftwosimilarpolygonsisequaltothesquareoftheratioofthecorrespondingsides.
Given:∆ABC~∆A’B’C’ Prove:∝∆
∝∆ ’ ’ ’,where
SASSimilarityTheorem:Ifanangleofonetriangleisequaltoanangleofanothertriangleandthesidesincludingtheseanglesareproportional,thenthetrianglesaresimilar.
Given:∆ABCand∆A’B’C’with∠A=∠A’and .
Prove:∆ABC~∆A’B’C’
SSSSimilarityTheorem:Ifthesidesofonetriangleareproportionaltothesidesofanothertriangle,thenthetrianglesaresimilar.
Given:∆ABCand∆A’B’C’with .
Prove:∆ABC~∆A’B’C’
Chapter11–TheRightTriangle
Theorem49:Thealtitudetothehypotenuseofarighttriangleformstwotrianglessimilartoitandtoeachother.
Corollary1toTheorem49:Thealtitudetothehypotenuseofarighttriangleisthegeometricmeanbetweenthesegmentsintowhichitdividesthehypotenuse.
Corollary2toTheorem49:Eachlegofarighttriangleisthegeometricmeanbetweenthehypotenuseanditsprojectiononthehypotenuse.
PythagoreanTheorem:Inarighttriangle,thesquareofthehypotenuseisequaltothesumofthesquaresofthelegs.
Theorem50–TheIsoscelesRightTriangleTheorem:Inanisoscelesrighttriangle,thehypotenuseis√2timesthelengthofaleg.
CorollarytoTheorem50:Eachdiagonalofasquareis√2timesthelengthofoneside.
Theorem51–The30° 60°RightTriangleTheoremIna30° 60°righttriangle,thehypotenuseistwicetheshorterlegandthelongerlegis√3timestheshorterleg.
CorollarytoTheorem51:Analtitudeofanequilateraltrianglehavingside is√ anditsareais√ .
Def:Thetangentofanacuteangleofarighttriangleistheratioofthelengthoftheoppositelegtothelengthoftheadjacentleg.
tanθ tangentofθoppositeadjacent
Def:Thesineofanacuteangleofarighttriangleistheratioofthelengthoftheoppositelegtothelengthofthehypotenuse.
sinθ sineofθopposite
hypotenuse
Def:Thecosineofanacuteangleofarighttriangleistheratioofthelengthoftheadjacentlegtothelengthofthehypotenuse.
cosθ cosineofθadjacent
hypotenuse
Theorem52:Twononverticallinesareparallelifftheirslopesareequal.
Theorem53:Twononverticallinesareperpendicularifftheproductoftheirslopesis‐1.
Theorem54–TheLawofSines:Ifthesidesopposite∠ , ∠ , ∠ of∆ havelengths , , ,thensin sin sin
Theorem55–TheLawofCosines:Ifthesidesopposite∠ , ∠ , ∠ of∆ havelengths , , ,then2 ∗ cos , 2 ∗ cos , 2 ∗ cos
Ch12–Circles
12.1‐Circles,Radii,andChords
Def:Acircleisthesetofallpointsinaplanethatareatagivendistancefromagivenpointintheplane.
Def:Circlesareconcentricifftheylieinthesameplaneandhavethesamecenter.
Def:Aradiusofacircleisalinesegmentthatconnectsthecenterofthecircletoanypointonit.Theradiusofacircleisthelengthofoneoftheselinesegments.
Corollary:Allradiiofacircleareequal.
Def:Achordofacircleisalinesegmentthatconnectstwopointsofthecircle.
Def:Adiameterofacircleisachordthatcontainsthecenter.Thediameterofacircleisthelengthofoneofthesechords.
Theorem56:Ifalinethroughthecenterofacircleisperpendiculartoachord,italsobisectsthechord.
Theorem57:Ifalinethroughthecenterofacirclebisectsachordthatisnotadiameter,itisalsoperpendiculartothechord.
Theorem58:Theperpendicularbisectorofachordofacirclecontainsthecenterofthecircle.
12.2‐Tangents
Def:Atangenttoacircleisalineintheplaneofthecirclethatintersectsthecircleinexactlyonepoint.
Theorem59:Ifalineistangenttoacircle,itisperpendiculartotheradiusdrawntothepointofcontact.
Theorem60:Ifalineisperpendiculartoaradiusatitsouterendpoint,itistangenttothecircle.
Theorem59:Ifalineistangenttoacircle,itisperpendiculartotheradiusdrawntothepointofcontact.
Theorem60:Ifalineisperpendiculartoaradiusatitsouterendpoint,itistangenttothecircle.
12.3‐CentralAnglesandArcs
Def:Acentralangleofacircleisananglewhosevertexisthecenterofthecircle.
Def:Areflexangleisananglewhosemeasureismorethan180o
Def:Thedegreemeasureofanarcisthemeasureofitscentralangle.
Theorem61:Inacircle,equalchordshaveequalarcs.
Theorem62:Inacircle,equalarcshaveequalchords.
12.4InscribedAngles
Def:Aninscribedangleisananglewhosevertexisonacircle,witheachoftheangle'ssidesintersectingthecircleinanotherpoint.
Theorem63:Aninscribedangleisequalinmeasuretohalfitsinterceptedarc.
Corollary1toTheorem63:Inscribedanglesthatinterceptthesamearcareequal.
Corollary2toTheorem63:Anangleinscribedinasemicircleisarightangle.
Theorem63:Aninscribedangleisequalinmeasuretohalfitsinterceptedarc.
Themeasureofacentralangleisthesameasthedegreemeasureofitsinterceptedarc.
Themeasureofaninscribedangleishalfthedegreemeasureofitsinterceptedarc.
Inscribedangleswiththesameinterceptedarcsareequal.
Anglesinscribedinsemicirclesarerightangles.
12.5SecantAngles
Def:Asecantisalinethatintersectsacircleintwopoints.
Def:Asecantangleisananglewhosesidesarecontainedintwosecantsofacirclesothateachsideintersectsthecircleinatleastonepointotherthantheangle'svertex.
Theorem64:Asecantanglewhosevertexisinsideacircleisequalinmeasuretohalfthesumofthearcsinterceptedbyitanditsverticalangle.
Theorem65:Asecantanglewhosevertexisoutsideacircleisequalinmeasuretohalfthedifferenceofitslargerandsmallerinterceptedarcs.
Themeasureofasecantanglewhosevertexisinsidethecircleishalfthesumofitsinterceptedarcs.
Themeasureofasecantanglewhosevertexisoutsidethecircleishalfthedifferenceofitsinterceptedarcs.
Alinethroughthecenterofacirclebisectsachordifthetwoareperpendicular.
Iftwochordsintersectinsideacircle,theproductofthetwosegmentsofonechordisequaltotheproductofthetwosegmentsoftheotherchord.
12.6‐TangentSegmentsandIntersectingChords
Def:Ifalineistangenttoacircle,thenanysegmentofthelinehavingthepointoftangencyasoneofitsendpointsisatangentsegmenttothecircle.
Theorem66:Thetangentsegmentstoacirclefromanexternalpointareequal.
Theorem67:TheIntersectingChordsTheorem
Iftwochordsintersectinacircle,theproductofthelengthsofthesegmentsofonechordisequaltotheproductofthelengthsofthesegmentsoftheotherchord.
Ch13–ConcurrenceTheorems
Ch13‐TheConcurrenceTheorems
13.1‐TrianglesandCircles
Def:Apolygoniscycliciffthereexistsacirclethatcontainsallofitsvertices.
Theorem68:EveryTriangleiscyclic.
Def:Apolygonisinscribedinacircleiffeachvertexofthepolygonliesonthecircle.Thecircleiscircumscribedaboutthepolygon.
CorollarytoTheorem68:Theperpendicularbisectorsofthesidesofatriangleareconcurrent.
Construction9:Tocircumscribeacircleaboutatriangle.
13.2‐CyclicQuadrilaterals
Theorem69:Aquadrilateraliscycliciffapairofitsoppositeanglesaresupplementary.
13.3‐Incircles
Def:Acircleisinscribedinapolygoniffeachsideofthepolygonistangenttothecircle.Thepolygoniscircumscribedaboutthecircle.Thecircleiscalledtheincircleofthepolygonanditscenteriscalledtheincenterofthepolygon.
Theorem70:Everytrianglehasanincircle.
CorollarytoTheorem70:Theanglebisectorsofatriangleareconcurrent.
Construction10:Toinscribeacircleinatriangle.
13.4‐TheCentroidofaTriangle
Def:Amedianofatriangleisalinesegmentthatjoinsavertextothemidpointoftheoppositeside.
Theorem71:Themediansofatriangleareconcurrent.
Def:Thecentroidofatriangleisthepointinwhichitsmediansareconcurrent.
Theorem72:Thelinescontainingthealtitudesofatriangleareconcurrent.
Def:Theorthocenterofatriangleisthepointinwhichthelinescontainingitsaltitudesareconcurrent.
13.5‐Ceva'sTheorem
Def:Acevianofatriangleisalinesegmentthatjoinsavertexofthetriangletoapointontheoppositeside.
Theorem73:Ceva'sTheorem
Thepointatwhichtheperpendicularbisectorsofthesidesofatriangleareconcurrentisthecenterofacirclecircumscribedaboutthetriangle.
Thepointatwhichtheanglesbisectorsofatriangleareconcurrentisthecenteroftheincircle,oracircleinscribedwithinthetriangle.
Thepointatwhichthemediansofatriangle,orlinesegmentsjoiningeachvertextothemidpointoftheoppositeside,areconcurrentisthecentroid,orcenterofmass.
Thepointatwhichthelinescontainingthealtitudesofatriangleareconcurrentistheorthocenter.
Ch14‐RegularPolygonsandtheCircle
14.1‐RegularPolygons
Def:Aregularpolygonisaconvexpolygonthatisbothequilateralandequiangular.
Theorem74:Everyregularpolygoniscyclic.
Def:Anapothemofaregularpolygonisaperpendicularlinesegmentfromitscentertooneofitssides.
14.2–ThePerimeterofaRegularPolygon
Theorem75–Theperimeterofaregularpolygonhaving sidesis2 ,inwhich sin and isitsradius.
Lengthofonesideofaregular ‐gonis2 sin
Perimeterofaregular ‐gonis2 sin
14.3–TheAreaofaRegularPolygon
Theorem76–Theareaofaregularpolygonhaving sidesis ,inwhich sin cos and isitsradius.
14.4–FromPolygonstoPi
Def:Thecircumferenceofacircleisthelimitoftheperimetersoftheinscribedregularpolygons.
Theorem77–Iftheradiusofacircleis ,itscircumferenceis2 .
CorollarytoTheorem77–Ifthediameterofacircleis ,itscircumferenceis .
14.5–TheAreaofaCircle
Def:Theareaofacircleisthelimitoftheareasoftheinscribedregularpolygons.
Theorem78–Iftheradiusofacircleis ,itsareais .
14.6–SectorsandArcs
Def:Asectorofacircleisaregionboundedbyanarcofthecircleandthetworadiitotheendpointsofthearc.
Ifasectorisacertainfractionofacircle,thenitsareaisthesamefractionofthecircle’sarea.Ifanarcisacertainfractionofacircle,thenitslengthisthesamefractionofthecircle’scircumference.
Ch15–GeometricSolids
15.1–LinesandPlanesinSpace
Postulate11–Iftwopointslieinaplane,thelinethatcontainsthemliesintheplane.
Postulate12–Iftwoplanesintersect,theyintersectinaline.
Def:Twolinesareskewifftheyarenotparallelanddonotintersect.
Def:Twoplanes,oralineandaplane,areparallelifftheydonotintersect.
Def:Alineandaplaneareperpendicularifftheyintersectandthelineisperpendiculartoeverylineintheplanethatpassesthroughthepointofintersection.
Def:Twoplanesareperpendiculariffoneplanecontainsalinethatisperpendiculartotheotherplane.
15.2–RectangularSolids
Def:Apolyhedronisasolidboundedbypartsofintersectingplanes.
Def:Arectangularsolidisapolyhedronthathassixrectangularfaces.
Theorem79–Thelengthofadiagonalofarectangularsolidwithdimensions , ,and is√ .
CorollarytoTheorem79–Thelengthofadiagonalofacubewithedgesoflength is √3.
15.3–Prisms
15.4–TheVolumeofaPrism
Geometry‐ReviewofCh12,14,&15
12
12
12
12
Θ
Α
2
2
2 2
43
4
13
13
13
cos cos
sin sin
2 .
cos sin
cos180
sin180
2
2 sin180
360
12
180
Ch16
Statement Euclid Lobachevsky RiemannThroughapointnotonaline,thereis
exactlyonelineparalleltotheline.
morethanonelineparalleltotheline.
nolineparalleltotheline.
ThesummitanglesofaSaccheriquadrilateralare
right. acute. obtuse.
