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)#MS%>#MS%J&-#PR%>#PR%LQ?F#GA%>#GA%]al[b#GV%>#GV%sww~#GP%>#GP%#GL%>#GL%Basic Geometrical Notation Disk 1$U.DEF.*.intro.*$U.EX.*.intro.?$U.REV.*.intro.S$U.DEF.*.circ.g$U.DEF.*.rect.{$U.DEF.*.tria.$U.DEF.*.dimen.$U.DEF.*.perspec.!"#$% $U$1.it.$1 introduction01$U* next *.GO.*.meas.IntroductionPress RETURN to continue or press HELPIntroduction to geometryExample of geometryExample of geometryDefinition of geometryPress RETURN to continue or press HELPPress RETURN to continue or press HELPDefinition of circlePress RETURN to continue or press HELPDefinition of rectanglePress RETURN to continue or press HELPDefinition of trianglePress RETURN to continue or press HELPDefinition of dimensionPress RETURN to continue or press HELPDefinition of perspectivePress RETURN to continue or press HELP=$U.DEF.*.meas.(@$U.EX.*.meas.<C$U.REV.*.meas.,-/$U$1.it.$1 measure+,$U* next *.GO.*.perim.'Measure+012347>)Test of measure$U$3+$U6 cm9$Usix cmK$U*" 6 cm "*^$U6 centimet*s$Usix centimet*$U*" 6 centimet"*$U*" 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"sq*unit*u$U4 + 6 + 5 = 15 sq*unit*$Ufifteen sq* unit*`#GA%^_Review of areaTest of area$U$3&Y$Uno/Y$Un@Y$U*" not "*QY$U*" NOT "*^b`#GA%Review of areaU$U.DEF.*.volum.*X$U.EX.*.volum.?[$U.APP.*.volum.T^$U.REV.*.volum.kgri$U$1.it.$1 volume*+$U* next *.GO.*.pnt.dVolumePress RETURN to continue or press HELPfjklmpqrs[b#GV%>#GV%wPress RETURN to continue or press HELPhpqrPress RETURN to continue or press HELPkpqrtxIntroduction to volumeDefinition of volumePress RETURN to continue or press HELPTest of volume$U$3-$U* yest*<$Uyes" "*K$U*" "yesV$Uyesi$U*" "yes" "*z$U*" "y" "*$Uyn#GV%oVolumeExample of volumeExample of volume Application of volumePress RETURN to continue or press HELP   Test of volume $U$3/$U* square*K$Utwenty seven cu* un*g$Utwenty-seven cu* un*y$U27 cu* un*$U*" "27 cu* un*n#GV%otuReview of volume $U$3 $U* square *4$U27 cu* unit*L$Utwenty seven cu*d$Utwenty-seven cu*x$U27" "cu* un*$U*" "27 cu* un*n#GV%ovVolumeTest of volume$U$30$U* square *D$U25 cu* unit*X$U25 cu* yard*i$U25 cu* 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continue or press HELP Test of vertex!$U$3'b$Ue6b$U*" e "*Eb$U*" E "*Rb$U*" "e_b$Ue" "*m#GP%Review of vertex"#$%t$U.DEF.*.seg.*w$U.DEF.*.collin.?z$U.DEF.*.concu.Q}$U.DEF.*.ra.d$U.DEF.*.par.vt$U.EX.*.seg.w$U.EX.*.collin.z$U.EX.*.concu.}$U.EX.*.ra.$U.EX.*.par.t$U.REV.*.seg.w$U.REV.*.collin.z$U.REV.*.concu.}$U.REV.*.ra.%$U.REV.*.par.8$U.DEF.*.lin.J$U.EX.*.lin.]$U.APP.*.lin.pq$U.REV.*.lin.$U$1.it.$1 line()$U* next *.GO.*.intro.LinePress RETURN to continue or press HELP\c#GL%>#GL%Press RETURN to continue or press HELP&Description of linePress RETURN to continue or press HELP'()*+,Application of linePress RETURN to continue or press HELP-Example of linePress RETURN to continue or press HELP./Definition of collinear pointsPress RETURN to continue or press HELP0Definition of concurrent linesPress RETURN to continue or press HELP1Review of line2Definition and example of line segmentPress RETURN to continue or press HELP3Definition and example of rayPress RETURN to continue or press HELP4Definition of parallel linesPress RETURN to continue or press HELP5Test of line6$U$3&L$Ub)3L$U*" "b<L$UbIL$Ub" "*W#GL%Review of line78Test of line9$U$3%$Uc2$Uc" "*<$Uc)I$U*" "c^$Uray * segments$Usegment * ray|$Ua$Uray$Ua)$Usegment$Ub$Ub)$Ud$Ud)$Uradius#GL%Test of line9$U$3%|$Uc2|$Uc" "*<|$Uc)I|$U*" "ca|$U* ray * segment*y|$U* segment * ray*#GL%:Review of line;Test of lineUse cursor control keys or mouse<>>$U$3GJJU#GL%Review of linePress RETURN to continue or press HELP=Test of lineUse cursor control keys or mouse<>>$U$3GJJU#GL%Press RETURN to continue or press HELPLine>Test of line?$U$3)h$Ub" "*6h$U*" "b?h$UbIh$Ub)Wh$U*" b)"eh$U*" B)"#GL%Review of line@Review of lineAReview of lineB?$U$3Z$Ub" "*(Z$U*" "b1Z$Ub;Z$Ub)IZ$U*" b)"WZ$U*" B)"z#GL%Test of lineC$U$3/D$Uconcurrent*AD$Uconcurent*O#GL%Review of lineDGBasic Geometrical NotationAnswer this question, and press RETURNPress RETURN to continue or press HELP*h*.**U*D#MS#GV#GP#GL#PR#GA&5<ANc{  Press RETURN to continue or press HELPPress RETURN to continue or press HELPPress RETURN to continue or press HELPReviewPress RETURN to continue, Review /9@HViPress RETURN for DESKTOPEBasic Geometrical Notation Disk 15D#MS%>AR#MS%>c$4masteryq$4needs workq$4incomplete}#PR%>#PR%>$6mastery$6needs work$6incomplete#GA%>#GA%>$7mastery$7needs work$7incompleteF$3Basic Geometrical Notation Disk 15D#GV%>AR#GV%>c$6masteryq$6needs workq$6incomplete}#GL%>#GL%>$5mastery$5needs work$5incomplete#GP%>#GP%>$7mastery$7needs work$7incomplete$3*HThe metric unit, and its3Habbreviation, referred to<Hin this course is:LHcentimetercmHYou must always use theHcorrect units in answeringZHRemember:cHThere is no period afterlHthe abbreviated metric unit.Hquestions.@HINTRODUCTIONJ-HHow long is each marked6Hsectionof the broken ruler?K"HYou can see from the markings onthe+Hruler that each section is 6 cm long.PH\$3L@HMEASUREH J@GPERIMETERJ GThe perimeter of the garden is the sumGof the length of its sides:Gp= 1 y + 2 y + 1 y + 2 y'Gp = 6 y[Gyards=GyMG1#G11VGThe formula to determine the_Gperimeter of ANY figure is:lGp = sum of length of sidesN HWhat is the perimeter Hofthis figure ?OHTo calculate the perimeter, youHneedto find the total lengthHof thesides of this pentagon.(Hp = 1.3 cm + 1.4 cm + 1.2 cm +0H+ 1.4 cm + 1.2 cmhHp = 6.5 cmpH\$3P ,HFind the length and width 5Hofa rectangle that would >Hhavethe same perimeter GHas thistriangle. PHGive the answer in whole YHnumbers.Q HThe perimeter of this triangle His12 units. HArectangle with the same #Hperimeterwould have sides of:H2 units and 4 unitsCHorLH1 unit and 5 units ,Heither:pH\$3UHor^H3 units and 3 units.R(HThe length of the sides of a1Hrectangleare 5.5 units andKHHow many square units will fitTHintothe rectangle ?:H4.O units.S+HTo calculate the area we multiply4Hthe length with the width:DH5.5 units x 4.O units, and thisMHgives us 22 square units.pH\$3@HAREAJ7HAre 2 square in. and 2-in. square@Hthe same ?SHA 2-in.square is a square with sides#Hof 2 in., and an area of 4 square in.3HA 2 square in. figure has sides of<H2 in. and 1 in., and an area ofEH2 square in.]HThe figures are NOT the same.pH\$3@HVOLUMEH J@HPOINT H J@HLINE H JGBasic Geometrical Notation G_!0GDISK1LGVolume \$6:G(cont'd)GSCOREXGPoint \$7dGLine \$5TUV 4GARRAKIS ADVANTAGELGDeveloped by UGARRAKIS TECHNOLOGIES tGCopyright 1986W GThe concepts coveredin this tutorial are presented in a predetermined order. (GYou may, however, interrupt the program AT ANY TIME to: @G* ask questions * test your progress * review a concept * advance to a new concept qGquestion and press RETURN. hGJust type your message orX GOVERVIEW DISK 10GINTRODUCTION]GAREATGPERIMETER9GMEASUREKG* Length of sides GHere is what you will learn on this disk:fGVOLUMEBG* Number of sidestG(cont'd)X(GPOINTCG* Collinear points:GLINELG* Concurrent lines^G* Ray&G 1G* VertexUG* Line segmentgG* Parallel lines GOVERVIEW(cont'd)DISK 1XGUSE THESE WORDS TO:G*Go to another concept:Ggoto, show me, teach(G*Review a concept:0Greview, review all@G*Test your progress:HGexam, testXG*Ask questions:define,`Gexample, what ispG*Other commands:score, overview,xGquit,again, restart, resumeYGGeometry has been used throughout history to solve problems -(Gfrom the planning of0GpyramidshGto navigation throughpGspace.ZGGeometry is one of the most widely applied branches of mathematics.8GAstronomy uses geometry to calculate the position of stars.[0GGeometry is used in9Gclothingdesign toBGcreate the shape andKGform of clothes.\ (GGeometry may be defined as a branch of mathematics that studies relationships, properties and measurements of points, lines, angles, surfaces, and solids.X(GCIRCLE:8GA closed plane curve where eachHGpoint is the same distance fromXGa fixed point within the curve.GRECTANGLE: GA closed plane figure having four0Gsides where the opposite sides@Gare parallel to each other andPGall angles measure 9OPI.`IA square is a rectangle whosepIfour sides are of equal length.@GA closed plane figure having0GTRIANGLE:PGthree sides.(GDIMENSION:8GThe measurement in a line ofHGlength, breadth, height,XGthickness or circumference.GPERSPECTIVE: GThe method of representing on a0Gplane or curved surface the space@Grelationships of objects as theyPGappear to the eye. The object`Gtherefore, seems to have length, pGwidth, and depth.;GWhen you have a figure or anDGobjectyou often need to MGdiscuss itscharacteristics.VGSo you measure it.GMeasure is one of the mostGbasic concepts in geometry.)G6]BHIn measuring a figure, you compareKHit against an accepted standardTHunit.^HBy measuring the number of miles&Hbetween two cities, you know how/Hfaryou have to travel from one 8Hpoint toanother._(GBy measuring the angle of a hill, you can determine if the hill is safe to ski.`GYou can measure different#Gcharacteristics of figures,Gand objects: ;G* perimeter FG* area QG* volume \G* size of angles TGThesides of a figure are the straightedges that form idGts boundary.aGThis figure has 8 sides.G1 G22G3CG64PG5 3G7"G8NGThis figure has 8 sides.G1 G22G3CG64PG5 3G7"G8`GThe front and back of the figure are not sides.tGThey are SURFACES.N GThe measure of the sidesGis knownas LENGTH.b G*The LENGTH of a sideGis the distance betweenthe beginning and endof that sidecGis the distance betweenthe beginning and endof that side 0G* It is measured with8Ga ruler, or other such instrument G* The LENGTH of a sidedGis the distance betweenthe beginning and endof that side 0G* It is measured with8Ga ruler, or other such instrument G* The LENGTH of a side PG* Length is always XGrepresented by a number and a unit, such as inches or meters.eGThe sum of the lengthofGthe sides of a figureis#Gknown as the PERIMETER.fGThe sum of the lengthofGthe sides of a figureis#Gknown as the PERIMETER.jGThe perimeter of this figure is 24 units.gGTo find the length of fenceneeded to enclose this garden, you must determine its perimeter.[Gyards=GyMGYou use the length of the sides to find the perimeter of a figure.DGYou use the length ofVGthe AREA of itsMGthe sides to determine_Gsurface, as well.h G* AREA is a measure of(Gthesurface of an objectc G* AREA is a measure of(Gthesurface of an ?G*Area is measured inGGsquareunits such as OG square inches0GobjectdGThe length of each of the four sides of this square is one unit. G10G11CG1iGThe length of each of the four sides of this square is one unit. G10G11CG1XGThe area ofthe square is one square unit, because each side isone unit long.iGThe one square unit is used tocalculate the area of any figure.0GTwo squares of one square unithave a combined area oftwosquare units.jGFinding the area of any surface means trying to determine how many square units will fit on it.kGFinding the area of any surface means trying to determine how many square units will fit on it.hGThis object has an area of 12 square units.k GTo find the area of any square or rectangle, just 0G3multiply of0Gthe length 8Gthe two adjacent sides.IG4l GTo find the area of any square or rectangle, just 0G3multiply of0Gthe length 8Gthe two adjacent sides.IG4 HGThe area of this rectangle is 12 square units. `GYou have multiplied the length by the width.m0GWhat is the area of this figure?)G2G2G25G3XG5IG1no GTo find the area, split the figure into simple rectangles.HGThen add the areas of each rectangle. The area of this figure is 15 square units.G2+G2(G_&G2 ;G3ZG5OG1pG\$3pq0GEvery figure has"08Ga surface area, but only three- dimensional objects have volume.r G*GVOLUME is a measure#GofGthe amount of space an object occupiesc G*GVOLUME is a measure#GofGthe amount of space an object occupies 8G*8GAnything that occupies space has volumed 8GDoes a flat piece of wood@Ghave volume?S8GSince a piece of wood has thickness, it occupiesspace and therefore has volume.pG\$3#GThrow a rock into a tank of water. Notice that the water level rises.s#GThrow a rock into a tank of water. Notice that the water level rises.KGThe rock has volume because it takes up space in the tank.t GVolume is measured in cubic units.@GThis cube has a volume of eight cubic units.u (GWe add square units to 8Ga surface.XGWe add cubic units to find theiGof an object. 0Gfind the area of a`Gfind volume0Gi1G_aG_vGTo calculate the volumeof a rectangular box, multiply the G by the  by theGlength width;GdepthOGwidthXGlengthG__"Gdepth.#G_wGTo calculate the volumeof a rectangular box, multiply the G by the  by thehGThe volume of this box is six cubic units(3x2x1).TG3PG2;G1GlengthwidthG__"Gdepth.#G_IG_w0GWhat is thevolume of Rubik's cube? (Assume that each block is one cubic unit.)x(GVolume is measured by determining the number of cubic units that fit into an object.PGThe volume of a cube is length x width x depth.GRemember:pG\$3(GThe volume of Rubik's cube is8G= length x width x depthHG= 3 x 3 x 3XG= 27 cubic unitspG\$38GYou want to add 1/2yard of topsoil to alawn of fifty squareyards.NG What volume ofNGtopsoil do you need?SpG\$3(GTo calculate volume you multiply the length by the width by the depth.HGThe area of an object is the length multiplied by the width.0GThe volume of topsoil neededis@G= length x width x depthPG= 5O x MG_ LG1 VG2`G= 25cubic yardspG\$3 Gare the building blocks of language, 0GPOINTSPGand PLANES@GLINEShGare the building blocks of geometry.GJust as letters, words, and ideasGFrom these building blocks, Gfigures such as GTRIANGLES,0GSQUARES,@GRECTANGLES,`GCIRCLESPGand pGcan be constructed. GA point cannot be defined. However, we can say that:X GA point cannot be defined. However, we can say that: G* A point is the smallest "Gelement in geometryc GA point cannot be defined. However, we can say that: G* A point is the smallest "Gelement in geometry 2G* A point has position in :Gspace and is represented by a dotd GA point cannot be defined. However, we can say that: G* A point is the smallest "Gelement in geometry 2G* A point has position in :Gspace and is represented by a dot RG* All figures are constructedZGed from points e GA point cannot be defined. However, we can say that: G* A point is the smallest "Gelement in geometry 2G* A point has position in :Gspace and is represented by a dot RG* All figures are constructedZGed from points jG* Points have neither length rGnor thicknessy GAlthough points themselves cannot be defined, air traffic controllers use them to define the position of airplanes.zhGThe point at which sides intersect is called a VERTEX.{XGMove the circle over a vertex of this figure.| GRemember: 0GA vertex is a point @Gwhere the sides of PGa figure MEET.`GThis figure has 4 vertices.,G24G1HG3}PGMove the circle overa point on the figure that is NOT avertex.~XGThese four points are not vertices. G148G23}GLook at the map below. Start at position A. Which position will you be at after following these directions?0GTurn left at the third vertex, then turn right at the second vertex and move forward two points.GThis is the correct path:xG\$3GThis is the correct path: GTurn left at the third vertex,xG\$3GThis is the correct path: GTurn left at the third vertex,8Gthen turn right at the second vertexxG\$3GThis is the correct path: GTurn left at the third vertex,8Gthen turn right at the second vertexXGand move forward two points.xG\$3GYou have completed$GDisk 1.5GYou can review theYGInsert the secondbGdiskand "reboot">Gmaterial on thisGGdisk or proceed toPGDisk 2.kGthemachine.X GA line, like a point, cannot be defined. However, we can say that lines:#$G LG lG c GA line, like a point, cannot be defined. However, we can say that lines: $G* Consist of a series of ,Gconnected pointsc GA line, like a point, cannot be defined. However, we can say that lines: $G* Consist of a series of ,Gconnected points 8G* Are straightd GA line, like a point, cannot be defined. However, we can say that lines: $G* Consist of a series of ,Gconnected points 8G* Are straight DG* Continue forever in both LGdirectionsd GA line, like a point, cannot be defined. However, we can say that lines: $G* Consist of a series of ,G, ,Gconnected points 8G* Are straight DG* Continue forever in both LGdirections XG* Have position in spacee GA line, like a point, cannot be defined. However, we can say that lines: $G* Consist of a series of ,G, ,Gconnected points 8G* Are straight DG* Continue forever in both LGdirections XG* Have position in space hG* Have length but no pGthicknessor widthypGLines can be used to show perspective.(GThis is how a line is represented.(GThis is how a line is represented.PGThe arrows indicate that the line extends forever in both directions.GPoints on the same line are COLLINEAR.XGLines that go through the same point  hGare CONCURRENT. XGCollinear points lie on the same line. GA LINE SEGMENT is part of a line. The line segment is defined by two points.hGand the section of line between the two points.`GIt includes the two pointsGA RAY is the part of a line that extends from a point in one direction. The ray includes the endpoint.0Gsame plane that never meet are calledPGPARALLEL LINES.(GLines in the GWhich of these figures represents a line? hGc)d) HGa)b) (GA line is represented by a segment with an arrow at each end.pG\$3GOnly figure b represents a line.dGc)d)@Ga)b)pG\$3GWhat is the best descriptionEGb) a line segment5Ga) a rayUGc) a ray containingmGd) a radius ]Ga line segmentGof this figure? 0GPartially correct.XGTry AgainpG\$3 0GA figure can represent @Gboth a ray a line@GandAG___ PGsegment.pG\$3GPoint to the line that is parallel to this one.GRemember: GParallel lines cannot intersect.hGParallel Not parallelPGThis line is parallel`Gto the original line.GWhich of these figures is a line segment? @GA line is straight.pG\$3pG\$3GA line extends forever in both directions, whereas a line segment is defined by two points on a line (its endpoints).`GFigure b represents a line  segment. 0GA line segment is part of a line. HGLines are straight.pG\$30GIf two or more lines go through the same point they are called_.SpG\$3 0GIf two or more lines go @Gthrough the same point PGthey are called `GCONCURRENT lines.@GTHE ENDGBasic Geometrical Notation G_!0GDISK1AGMeasure \$4pG(cont'd)GSCOREMGPerimeter\$6YGArea \$7Tl|88||8xx8||8|<<`<|fl0fF8l8pvp88pf< 80 0 ~~`0 0```nff>fff~fff<<f<flxpxlf`~fv~~nfff<ff<f>``|fff|<```<>fff>ff>|``|f8<p``flxlf8<|ffff>|f```>`<|~f>fff<|lf<|~ 0~0p p` 44b~ 00||l|88||8||8||8|<<><~~< ~~ xx00<~~< 0``0$ff$<~~< fffllll>`<|fl0fF8l8pvp88pf< 80 0 ~~`0 0```nff>fff~fff<<f<flxpxlf`~fv~~nfff<ff<f>``|fff|<```<>fff>ff>|``|f8<p``flxlf8<|ffff>|f```>`<|~f>fff<|lf<|~ 0~0p p` 44b~ 0HH0~$ff~f&$,4$d<<f ~Ù}xc8ll8x~~~ `` 8||8  $ff$<~~< 888llllllll0|x 00f8l8vv``0```0`00`f<UA` a!bTc6UA`abTc.UA`ab Tc(UA`abTc"UAabTc UAabTcUAabTcUAA0` A0`A@#0`>2b>5e4'A@4d0Ad0Ad0Ad0Ad0Ad0Ad0Ad0Ad0A@1a@Aa>Aa< A0`>1a:2b>3c< 4d05e4'6f7gA@3c<$A@ƀ0`;1a42b2R3c4d5e6f7gA2A0`:2A2A1aA0`A0`%Aāłƃ0`%A@1a'A1aAaAa!A1a'A1a%2bAb Ab"!A2b 'A2b"3c"%Ab$Ab&!A2b$'A2b&4d&%Ab(Ab*!A2b('A2b*5e*%Ab,Ab.!A2b,'A2b.6f.%Ab0Ab2A2b0'A2b2!A2b0A2Aȁɂʃ˄̅͆·0`2k1a8^2b3cp4d$V5er6f7grA@`0fa6^2b8}cv4d$W5eA@`.aa4^b8x3cr4d$V7gA@`,_a2^b8r4d"WA@`*^a0^b6j4d$VA@`(^a.^b4b4d$WA@`&^a,^b2Z4d$VA@`$^a*^b0R4d"WA@`"^a(^b.O4d$VA@` ^a&^b,O4d$WA@`^a$^b*O4d$VA@`_a"^b(O4d"WA@`aa ^b&P4d$VA@`da^b$R3c8~4d$W7grA@`ka^b"Sc8{4d$VA@`oa^b Uc8w4d"WA@`ta^bXc8q4d$V5e8{A@`za^b[c8l4d$We8wfA@0`aab`c8g4d$Ve6rfA@adbfc8c4d"We4nA@$Ёт҃ӄԅ0`1a2b3c.4d5e>O6f7gA@0`&G3c4d. 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