-[UNDEFINEDTerms] [11Point; notation PointA is labeled wi th
acapi tal letter,A in thi s case[2JLine; notation Line KM is
labeledcitherKM or MK orli ne I [3JPlane; notation PlaneN is
labeledeit herplane 11 orpl aneABC ifpointsA, B, andC are on plane
/I [DEFINEDTerms] [GENERAL Terms][1J i( C rlCJI Jl" ) Shapes
arethesame shapeand size[21 (II 11"1' Shapesarcthesameshape,
butcanbedifferentsizes[3J i.qJdl Sets of po ints or numerica
lmeasurements arc exactly t he same[4J JIllor! Describesthe
resultwhen all ofthe pointsare puttogether[5J (II rs C 101)
Describes the pointswhere indicated shapes touch[6J p c The
setofall points111"'4'. [1J Collinear pointsareon the same line[2J
Non colill cI points are not on thesame line[3J Inter II 9 lines
have one and on lyonepointin common[4J PIp Idlcul, r lines
intersect and form90angles atthe intersection; 1.[5J k w lines are
not in the same plane,never touch, and go In differentdirections[61
Tlan versaI lines intersect two or moreco-pl anarlinesatdiffere nt
points[7J P r II I lines are co-planar (in the same .plane) , share
no points in common, donot intersect , go in the same directionand
nevertouch; II[LINESegments][1J The set ofany 2 points on a line
and allofthe collinear points between them;ABwhereAandB are the
endpoi nts ofthe Iinesegment[2J The I CJth is the distance between
the 2endpoints; it is a numerical va lue; ABmeans the length ofA
B1'.'.[1J The set ofcollinea r point s going in onedirection from
onc point (the endpoi nt oftheray) on a li ne ;notati on: ASwhereA
istheendpoint; not ice lIB *BAbecausetheyhave different endpoint s
and conta indifferent points on the li ne[2JOppo It r y
arecollinear,shareonlyacommon endpoin t and go in oppos it edirecti
ons[ANGLES] [1J The union oftwo rays thatshare one andonly one
point, the endpoint ofthe raysa. The Sid ofthe angl e are the rays
andthev r x is the endpointoftheraysb. The Interrol is all t he
points betweenthe two sides ofthe anglec.LABewhere B is thevertex
orsimply IB ifthere is onl y one angl e with vertex B[2J Ov
Ilapping angl s share somecommon int erior points[3J An ac Jte
angle measures less than 90[4JAn obtuse angle measures more
than90[5J A right angle measures exactly 90;it isindicated on
diagrams by drawing asquare in the corner by the vertex of
theangle[6] A straightangle measuresexactly 180[7J Complem ntary
ngles are two angleswhose measures total 90[8J Supplementary angles
are two angleswhose measurestotal 1800[9J Vertical angl s are
twoangles that shareo nly a eommon vertex and whose s idesform
lines[10JAdjacent angles are two angles t hatsha re exaetly one
vertex and one side,but no common interior points ; i.e., theydo
notoverlap[11JAn ngle bisector is a ray ora line thatcontains the
vertexofthe angle, is in theinterior, and separates the angle into
twoadjacent angles with equal measures[3] The n idpoint is a point
exactl y in themiddle ofthetwo endpoints[4] The bisector intersect
s a line segmentat its midpoint[5J The perpendicular bls tor
intersectsa linesegment at its midpointand forms90angl es at the
intersection1[TRANSVERSALLINE Angles][1JIn erlor ilngll are formed
with thc raysfrom the 2 li nes and the transversal,suchthat the
interiorregions ofthe anglesarelocated between the 2 lines[2J Alt
rn 1 eII trior mgl arc inter ior cangles wit h di fferent vertexes
andinterior regions on opposite sides oft hetransversal[3J ,1m lel
II t nor ngle are inter iorangl es wi th different vertexes an
dinterior regions on th e same s ide of thet ransversal[4JEx I
lor,r gl are formed with raysfrom the 2 linesand the transversal,
sucht hat thc in terior regions ofthe angles arenot bet ween t he 2
lines[51Altern t xt riar lnql are exteriorangles with different
vertexes andinterior regions on opposite sides ofthe iIIr..tra
nsversa l ,..[6] Carr pondlng angl have different =vertexes ; the
ir intcriorreg ionsare on thc ...same side ofthe transversa l and
in t he "same positions relative to the lines and mthe transversal;
one of th e pair of Zcorrespondingangles is an interi or anglcand
the other is an exteriorangle [POLYGONS] [1JPolygons arc plan
(flat), closed s hapesthat are formed by l ine segments tha tinter
ect only atthcirendpointsa. Not Thcy are named by listing
theendpoints of the l ine segments inorder, going either clockwise
orcountercl oc kwisc, sta rt ing at anyoneofthe endpointsb.The
sidesare line egmentsc. The int riar is all of t he pointsenclosed
by the s idesd.The xterior is all of the point on theplane ofthe
polygon, but neithe r ont he si des nor in t he int eriore.The
vertic (or vertexes) are the iIIr..endpoi nts ofthe li ne segments
,..f.Inc ludc a llthe pointson thes ides( line =segment s) and the
vertices ...g.The int rior angl or a polygon have "the same vert
ices as the verticcs ofthe mpolygon, have sides that cont ain the
Zsides o r the polygon, and have in teri orregions that conta in
the interior ofthe polygon- every pol ygon has as
manyinteriorangles as ithas verticesPolygons(continued)
h.Consecutive interior angles havevert ices that are endpoints
ofthe samesideofthe polygonI.The exterior angles are formed whenthe
sides of the polygon are extended;each has a vertex and one si de
that arealso a vertex and contain one side ofthe pol ygon; the
second si de of theexterior ang le is the extension
oftheotherpolygon sidecontainingtheanglevertex; the
interioroftheexteriorangleis part of the exterior region of
thepolygon; exteri or angles are s upp le-ments oftheiradjacent
interioranglesj.Diagonals of a polygon are linesegments wit h
endpoints that arevertices of the polygon, but thedi agonals are
notsides ofthe polygon[2) CONCAVEpolygons have at least
oneinteriorangle measuring morethan 1800[3) CONVEXpolygon s have no
inter iorangles more than 1800and all interiorangleseach measure
less than 180[4) REGULARpolygons haveall si de lengthsequal and a
ll interi or angl e meas uresequal[5) CLASSIFICATIONSOFPOLYGONS a.
Classifi ed by the number of si de s;eq ualto the nu mber of
verticesb. side lengths and angle measuresarc not necessa rily
equal un les s theword " regular" is also used to name thepolygonc.
Categories Triangles havethree sides Quadrilaterals have four sides
Pentagons have five sides Hexagons have six si des Heptagons have
seven sides Octagons havee ight sides Nonagons have nine si des
Decagons have ten si des n- gons have n s ides[CIRCLES] [1) The set
ofpoints in a pla ne eq uid is tan tfrom the center ofthe ci rcle,
which liesin the inter ior ofthe circl e and is not apointon the
circl e; 360[2) A radius is a linesegmentwhoseendpointsare thecent
erofthecircleand any poi ntonthe ci rcle; the length of a radius is
thedistanceofeach pointfrom thecenter[3) A chord is a li ne segment
whoseendpoints arc 2 points on the ci rcle[4] A diameter is a chord
that contains thecenterofthe ci rcle; the lengthofadi ameterIS the
distance from one pointto another on the circle, going
throughthecenter15) A secant is a li ne intersect ing a circle
intwo poi nts [6) SPECIALPOLYGONS a.Triangles Polygons with 3 sides
and 3 vertices;the symbol for a triangle is tri angleABC is written
An altitude (height) is a line segmentwith a vertex ofthe triangle
as oneendpoint and the point on the linecontaining the opposite
side of t hetriangle where the altitude is perpen-diculartothat
line; everytriangl e has 3alt itudes A base is a side ofthe
triangle on theline perpendicular to an altitude; everytriangle has
3 bases Formulaforarea A =tllb or iI=thb wherea=altitude,b=base
orwhere h=hei ght (altitude), b=basein2 ways,bysidelengths and by
anglemeasurements a]When classified byside lengths:Scalene have no
side lengths=, Isosceles have at least 2 sidelengths equal,
Equilateral have all 3 side lengthsequal; note it is also an
isoscel estriangleb]When classified byangle measure-ments:Obtuse
have exactly one anglemeasurement more than 900 Right have exact ly
one anglemeasurementequal to 9090 Acute have all 3 angles less
than0; note that if all 3 angles areequal, then t he tr iangle is
calledequiangularIsosceles triangles a]The vertex angle has s
idescontain ing the two congruent s idesofthe triangleb]The base is
the side with a differentlength than the other two sides; not[6]
Atangentisa line thatis co-planarwith acircle and intersects it at
one point only,call ed the pointoftangencya.A cornmon tangent is a
line that istangentto 2 co-planarcircles Common internal anqents
intersectbetween the twocircles Common external tangents do
notintersectbetween the circl esb .Two circ les are tangent when
they areco-planar and share the same tangentline atthe same
pointoftangency; theymaybeexternallyorinternally tangent[7) Equal
Circles haveequal-length radii[8) Concentric circles lie in the
same planeand have the samecenter2necessarily the side on t he
bottomofthe triangl ec] The base angles ofan isoscelestriangle have
t he base contai ned inone oftheir sides ; they are alwaysequal in
measureRight Triangles a] The hypotenuse is opposi tethe ri ght
angle and is the longestsideb ]The legs arc the 2 sides that arcnot
the hypoten use; the linesegments contained in the sides o ft he
right angleb.Quadrilaterals 4-sided polygons Have 2 diagonal s and
4 vert icesrap zOld have exactl y one pa ir ofpara ll e l sides ;
there is never moret han onepairofpara ll e lsidesa] Para ll el
sides : bab]Non-para llel sides :legsc] The 2 angles wi th vertices
that arcthe endpoint s o f the same base arccall ed ba
angld]lsosceles trapeZOids havelegs that arc the same length
Parallelograms have 2 pairs ofparallel s idesa] Rectangl s have 4
ri ghtanglesb]Rhombus s (s ing. rhombus) have4 sides equa l in
lengthc] Squares have 4 equal sides and 4equal angles ; therefore,
eve rysquare is both a recta ngle and arhombus[9] An inscribed
polygon has vertices thatare po ints on t he circl e; in th is sa
ill esi tuation, t he ci rcl e is c ircumscribed aboutthe
polygon110] A circumscribed polygor has sides thatarc segments of
tangents to theci rcle; i.e.,the s ides of the polygon each
containexac t ly one poi nt on the circle; in th issame sit uati
on, the circle is inscribed inthe polygon[11) Anarc is partofaci
rclea.A erni ircl is a n arc whoseendpoints are the endpo ints o f
adiameter; 180; exactl y th ree pointsmust be u.' cd to na me a
scmiei rcle;notation: AiJC where A and C are t heendpoints o fthe
diameterThrough a point not on a line, exactlyone
perpendicularcanbe drawn to thelineThe shortest dist ance from
anypoint to a line or to a plane is thepcrpcndiculardistanceThrough
a poi nt not on a line, exactlyone parallel can bedrawn to the
lineParallel lines are everywhere the samedistanccapartIfthree or
more parallel lines cut offequal segments on one transversal,
thenthcy cut off cqual segmcnt s on cvcrytransversal they sharcA
linc anda planc are parallel iftheydonot touch orintcrsectTwo or
more planes are parallel if theydo nottouch
orintersectAnglesaremeasured usinga protractoranddegree
mcasurements: There are 3600in acircle; placing the center ofa
protractor atthe vcrtcx ofan anglc and counting thedegreemeasure is
li ke puttingthe vertexofthe angle at the center of a circle
anc!comparing the angle measure to thedegreesofthecircle9()1 00If
two angl es are compl ements of thesame angl e, then they arc equal
Inmeasure (congruent)If two angles arc compl ements
ofcongruentangles, then theyare congruentIf two angl es are
supplements of thesameangle,then they are congruentb .A minor arc
length is less than thelengt h of t he semicircl e; only twopoints
maybe used to name a minor'arc ;notation: DEwhere D and E
aretheendpoints ofthe arcc.A majorarc lengt h is more than
thelength of the semi c irc le; exac t lythree points arc used t o
namc amajor arc; notation: FCfjwhere Fand H are the endpoints ofthe
arc[12JAcentral angle vertex is the cent erofthe c ircle with si
des that containradiiofthe circle[13JA inscribed-angle vert ex is
on acircle with sides that contain chordsofthe circleIf two
parallcl plancs arc bothintcrsccted by a third plane, thcn
thclinesofintersection are parallelIf a point lies on the
perpendicularbisectorofa line segment,thenthepointis equidistant
(equal distances) from thecndpoints ofthe line segmentIf a point is
equidistant from t heendpoints ofa line segment , then thcpoint
lies on thc perpendicular biscctorofthe line segmentTo trisect a
line segment, separate itinto three other congruent (equal
inlength) line segments, suchthatthe sumofthe lengths ofthe three
segments isequal to the length ofthe original linesegmentIf two
angles are supplements ofcongruentangles, then theyarecongruentVert
ical angles are congruent and haveequal measuresIfa point lies
onthebisectorofan angle,then the point is (equaldistances)from the
sidesofthe angleDistance from a point to a lineis always the length
of the perpen-dicular line segment that has the poi ntas
oneendpoint and a pointon the lineas the otherIfa point is
equidistant from the sides ofan angle, then the point lies on
thebiseetorofthe angleAn angle is trisected by rays or lines
thatcontainthe vertexofthe angleandseparatethe angle into three
adjacent angles (inpairs)thatall haveequalmeasures[POSTULATES]
Statements that have been acceptedwithoutformal proof[1J A
linecontainsat least2points,andany2points locate exactlyone line[2J
Any 3 non-collinearpoints locate exactlyoneplane[3] A
lineandonepointnotonthe line locateexactlyone plane[4J Any3 points
locateat leastoneplane[5J If 2 pointsofa lineare in a
plane,thentheline is in the plane[6] If2 points are in a plane,
then the linecontainingthe 2pointsis also in the plane[7] If2
planes intersect, then the intersectionis a line3If two rays do not
int ersect , then theunion of the rays is s imply a ll of thepoints
on both raysIftwo rays intersect in one and only onepoint, but not
at the endpo int, then theunion is all ofthe points on both
rays;theintersection is that one point where theytouchIftwo rays
intersect in one and onl y onepoint, the endpoint, then the uni on
is anangle; the int ersection is the endpointIft wo rays intersec t
In more than onepoint, then the union IS a line; theintersection is
a line segmentA BAB BAAB BAIf lines are para ll e l, th en th e a
lt erna teinter ior angl es of a tra nsve rsa l arecongruentIf the
a lt ern ate interi or angles of atransversal are congruent. then
the linesare para ll elIflines are para ll e l, t hen the same s
idcinteri or angl es of a trans ve rsa l aresuppl ementaryIf th e
sa me-s ide int e ri or angl es of atrans versal are s uppl eme
ntary, the n theI inesare para ll e lIf lines are para ll e l, then
thecorrespond ing angles ofa transversalarecongruentIf the corres
ponding ang les o f a trans-ve rs a l are congruent, then t he
lines areparall elIf lines a re parall e l, then th e a lte rn
ateexterior angles of a t ransversal arecongruentIf the al ternat e
exte rior ang les o f atransversa l are congruent, then the
linesare parallelIfa transversal is perpendi cular to one o ftwo
parall el lines, then it is al so perpen-----ndiculartothe
other------m alternat Jr.: 4- 6; 5- 3same-Side interior 4- 5; 3-
6correspondng':s: 1- 5; 4- 8;3- 7; 2- 6alternateexterior 1- 7;
2-8ductrnalThe sum ofthe measures ofthe interior '1The 3 bisectors
of the angles of aWhen an altitude is drawn to
theanglesofaconvexpolygonwith nsides is triangle intersect in
onepoint, which ishypotenuseofa righttriangle(n-2)180 degrees
equidistantfromthe 3sidesThe two triangles formed areTo find the
measureofeach interior The ofthesimilarto each otherand to
theangleofaregularpolygon,findthe sumof sides of a triangle
intersect in oneoriginal right triangleall ofthe interiorangles and
dividebythe point,equidistantfrom the 3verticesThe altitude is
thenumber of interior angles, thus, the The medians (line segments
whosebetween the lengths offormula (TI- 2) 180
endpointsareonevertexofthe trianglethe two segments of theTIand the
midpoint ofthe side oppositehypotenuseThe sum ofthe measures ofthe
exteriorthat vertex) ofa triangle intersect inEach leg is the
geometric meanangles ofany convex polygon, using oneone point
two-thirds ofthe distancebetweenthe hypotenuseand
theexteriorangleateachvertex, is 360fromeachvertex to
themidpointofthe sumlength of the segmcnt oftheIroppositeside
hypotenuse adjacent (touches) The3-angletotal
measurement=180Iftwosidesofatriangleare unequal inmberIf two angle
measurements of onelength, then the opposite angles arcto the
legidcstriangle=two angle measurements ofunequal
andthelargerangleisoppositet .anothertriangle,thenthemeasurementsto
the longer side; and conversely, ifIfthree sides
ofofthethirdanglesarealso=two angles ofa triangle arc unequal ,one
triangle are congruentto three Each angle ofan
thenthesidesoppositethoseanglesaresides ofanother, then the
triangles is 60unequaland the longerside is opposite are congruent
Therecan be no more than one rightorthe largerangle I f two sides
and obtuseangle in anyonetriangleThesumof thelengthsof anytwosides
the included angle ofone triangle The acute angles ofa right
triangle arcis greaterthan the length ofthe third are congruent to
two sides and the complementaryside; the difference ofthe lengths
ofincluded angleofanother. then the Themeasurementofan
exteriorangle=any two sides is less than the length oftrianglesare
congruent thesumofthemeasurementsofthetwothe thirdsideIftwo angles
andremote (not having the same vertex asthe included side ofone
triangle theexteriorangle) interioranglesAnequilateraltriangleis
alsoequian-are congruent to two angl es andI
ftwosidesofatriangleareequal ,thengular; and, an equiangular
triangle isthe included side ofanother, thenthe angles opposite to
those sides arealso equilateralthetriangles arecongruentalsoequal;
and, iftwo anglesareequal ,An equilateral triangle has three
60-Iftwoanglesandathen the sides opposite those angles
aredegreeanglesnon-included side ofone
trianglealsoequalThebisectorofthevertexangle ofanare congruent to
the twoIftwo sidesisoscelestriangle is the
perpendicularcorresponding angles and non-ofone triangle are equal
in length tobisectorofthebaseofthetriangle included side ofanother,
then the two sides ofanother, but the included trianglesare
congruent angle ofone triangle is largerthan theIn a right Ifthe
hypotcnuse included angle ofthe other triangle,triangle, , where .,
and ' and one leg ofa right triangle are then the longer third side
of thearethe lengthsofthe legsand is the triangles is opposite the
largerincludedlengthofthe hypotenusecongruent to the hypotenuse and
the corresponding leg ofanother.angleofthetrianglesIfthe square
ofthe hypotenuse isIftwo sidesequal to thesumofthesquaresofthethen
the two right triangles are ofonetrianglearc equal to
twosidesofothertwo sides, then the triangle is acongruent another,
but the third side ofone is longer than the third side ofthe
other,Ifthe square ofthe longest side is Iftwo then the
largerincludedangle(includedgreaterthanthesumofthesquaresof angles
of one triangle arc between the two equal sides) isthe other two
sides, then it is an congruent to two angles of opposite to the
longer third side ofthetriangle; ifit is less than theanother. then
the triangles are trianglessum ofthe squares ofthe othertwosimilar
(same shape but not sides,then it is an trianglenecessarilythe
samesi ze) If a line is parallel to one side andIn a 45-45-90 If
theintersects the other two sides, then ittriangle, the legs have
equal lengthssides ofone triangle are propor-divides those two
sides proportionally,and the length ofthe hypotenuse
istionaltothecorrespondingsidesof and creates2
similartriangles12timesthe length ofoneofthe legs another, then the
triangles areI fa anangleofatriangle,itIn a
30-60-90similardividesthe opposite side into segmentstriangle,the
length ofthe shortestlegIftwoproportional to theothertwosidesis 1/2
the length ofthe hypotenuse,sides ofone tri angle are propor-The
line segment that joins theandthe lengthofthe longerleg is 13
tional to two sides ofanother andmidpointsoftwo sidesofa triangle
hastimesthe length oftheshortestleg the included anglcs of each two
properties:The midpoint ofthe hypotenuse ofatriangle are congruent,
then theIt is to the third side,andright triangle is equidistant
from thetriangl esare similarIt is ofthethird
sidethreevertices4I
ATERALS 4The (the line segmentwhose endpoints are the midpointsZ
ofthe2 non-parallel sides) is parallelIII tothebases,andits length
is equal tohalfthe sum ofthe lengths ofthe 2D-basesC The may be
calculated byaveragingthe lengthofthe basesandT mUltiplying by the
height (altitudeothat is the length ofthe line segmentP that forms
90-degree angles with thed bases); thus , theformula:A(hi(hi +h, )h
et where the 2bases are b1 and b2 andthe height is h Two angles
with vertices that are theendpoints of the same leg of atrapezoid
areAll 4 interior angle measures ofalltrapezoids total 3600Thebase
anglesare congruent(has congruentlegs) Opposite angles are suppl
e-mentary Opposite sides are parallel andcongruentOpposite angles
arc congruentAll 4 interiorangles total 3600Consecutive interior an
g les (thei rvertic es are endpoint s for the sameside) arc suppl
ementaryDi agonal s bi secteach otherA quadril ateral is a
parallelogram if:[CIRCLES] Ifa line is to a circle, then itis
perpendicular to the radius whoseendpoint i s the point oftangency
(thepoint where the tangent line intersectsthe circle)/.If t wo
tangents to the same circleintersect in the exteriorregion, then
theline segments whose endpoints are thepointofintersection ofthe
tangentlinesandthetwo pointsoftangencyareequalin length; or, line
segments drawnfrom a co-planar exterior point of acircle to pointso
ftangencyon thecircleare congruentOne pair of opposite sides
iscongruentand parallelBoth pairs ofopposite sides arecongruentBoth
pairs of opposite anglesare congruentThe diagonals
bisecteachotherThe can be calculated bymultiplying the base and
theheight; that is, A=bh=hb Since opposite sides arebothparallel
andequal, any sidecan be the base; the height(altitude) is any line
segmentperpendicularto the base whoseendpoints arc on the base
andthe sideoppositethe baseParallelograms with 4
rightanglesDiagonals are congruent andbisecteachotherThe equal s
Iworhb where 1= length, w=width,h=height, and b =baseIf the4
sidesare all equal,thenthe rectangle is more specifi-callycalled a
squareParallelograms with 4 congruentsidesOpposite angles are
congruentAll 4 angle measures total3600 Any 2consecuti ve angles
arcsupplementaryIf a line in the plane of a circl e isperpendicular
to a radius at its outerendpoint. then the line is tangent to
thecircleThe measure ofa is equal tothe measureofits central
angleThemeasure ofa is 1800The measure ofa is equal to3600mlllus
the measure of itscorrespondingminorarcIn the same circle or in
equal circl es,equa I chords have equal arcs and equalarcs have
equal chordsA perpendicular to a chordbisectsthe chordand its arcIn
the same circle or in equal circles,congruent chords are the same
distancefrom the center, and chords the samedistance from
thecenterarccongruent5If4 int erior angles eachequal 900, th e n
therhombus is more specifi-cally called a squareThe diagonals
arcperpendicular bi sectorsofeach otherEach diagonal bi sects
thepair of oppos ite angl eswhose vertices arc th eendpoints ofthe
diagonal4 equal s ides and 4 equalang les; every squareboth a
rectangle and arhombusThe diagonals arecongruent, bi sec t
eachother, arc perpe ndicul a rto each other and bisectthe int
erioranglesA FThi s indicatethe re lati ons hips of quadri lateral
sA=QuadrilateralsB=RhombiC=RectanglesD=SquaresE=Trape
zoidsF=ParallelogramsAn isequal to hal fof its intercepted arc (the
arc whichli es in the interior ofthe insc ri bedangle and whose
endpoint are onthesidesoftheangle)r-... mMPNmMN If two int
erceptthe same are, then th e angles arecongruent] fa is inscribed
in acircle, t hen oppos ite ang les areupplemen taryAn angle insc
ri bed in a el11i eircleis always a ri ght angl
eAnCircles(coi/lilllled) An angle formed by a and a equal to half
the difference of the itse xternal segmentlength=the productis
equal to halfofthe measure interceptedarcs of the other secant and
its e xternalofits intercepted arc When two chords intersect inside
a segment lengthangle formed by two chords circle, the product of
the segment Whe n a tangent and a secant lineintersecting inside a
circle = to halfthelengths ofone chord=to the product of segment
are dra wn to a c ircle frol11 thesum ofthe intercepted arcsthe
segment lengthsofthe otherchord same exterior point, the square
ofAn angle formed by two secants, or twoWhen two secant line
segments are the length of the tangenttangents, or a secant and a
tangent, thatdrawn to a circle from the same exteriorsegment= to
the product o f th eintersectata pointoutsideofthecircleisendpoint,
the product ofone secant andsecantand its externa lsegment
lengthiUr-----------------------------------------------------------------Aa
The area, A, of a two-di mens ional shape is the number ofsquare
units that can be put in the region enclosed by the sidesArea is
obtained through some combination ofmultiplyinghei ghts and bases,
which always form 90 angles with each ot her,except in
circlesIfb=8,then: A=64square units A=lrh, orA=/w IfII =4and b= 12,
then: A =(4)(12),A=48 square units A=!hh ZA = 1(8)(12),A=48
squareunitsiUA=hhD.Ifh=6 and h=9, then: CA= (6)(9),A= 54sq uare
units A=1h(h,+h2)If h = 9, hi= 8 andh2=12, then:A=1(9)(8+12),A =
1(9)(20),A=90 square unitsA=lrt2 A=1rr2; If r =5,then:
A=n:52=(3.14)25=78.5 square units bl
C=21tr Ifr=5,then: C=(2)(3. l4)(5)=10(3. 14) = 3 1.4 units
blegsa and h,then: c2=a2+h2If a right triangle has
hypotenusecandV=/wlr If/ =12, w= 3 and Ir = 4,then:"o;pZ
V=(12)(3)(4), V=144 cub ic units w I "OTE: TO Due IU il sfonnal .
pl cn'>CD. fl .."igncu orl.;1\11 r ighl\ r escn ecl. No pan or
pll hli..: ation may herc p rt K! u
