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PointsLinesandPlanes
Unit1Lesson2
PointsLinesandPlanes
Studentswillbeableto:
•Drawpoints,lines,linesegments,rays,andplanes.• Identifypoints,lines,linesegments,raysand
planes.Knowprecisedefinitionsofline,andlinesegment,based
ontheundefinednotionsofpoint,line.
PointsLinesandPlanes
KeyVocabulary:Aline,Apoint,
Alinesegment,Aplane,
Intersection.
PointsLinesandPlanes
• Ingeometry,somewords,suchaspoint,line,andplane,areundefinedterms.Althoughthesewordsarenotformallydefined,itisimportanttohavegeneralagreementaboutwhateachwordmeans.
• Apointhasnodimension.• Itisusuallyrepresentedbyasmalldotandnamedbyacapitalletter.
PointsLinesandPlanes
• Aline extendsinonedimension.• Itisusuallyrepresentedbyastraightlinewithtwoarrowheadstoindicatethatthelineextendswithoutendintwodirections,andisnamedbytwopointsonthelineoralowercasescriptletter.
PointsLinesandPlanes
• Aplane extendsintwodimensions.• Itisusuallyrepresentedbyashapethatlookslikeatabletoporwall.• Youmustimaginethattheplaneextendswithoutend,eventhoughthedrawingofaplaneappearstohaveedges,andisnamedbyacapitalscriptletteror3non-collinearpoints.
PointsLinesandPlanes
• Alinesegment isasetofpointsandhasaspecificlengthi.e.itdoesnotextendindefinitely.• Ithasnothicknessorwidth,isusuallyrepresentedbyastraightlinewithnoarrowheadstoindicatethatithasafixedlength,andisnamedbytwopointsonthelinesegmentwithalinesegmentsymbolabovetheletters.
PointsLinesandPlanes
• Aray isasetofpointsandextendsinonedimensioninonedirection(notintwodirections).Ithasnothicknessorwidth,isusuallyrepresentedbyastraightlinewithonearrowheadtoindicatethatitextendswithoutendinthedirectionofthearrowhead,andisnamedbytwopointsontheraywitharaysymbolabovetheletters
PointsLinesandPlanes
• Collinearpoints arepointsthatlieonthesameline.
• Coplanarpoints arepointsthatlieonthesameplane.
PointsLinesandPlanes
SampleProblem1: Usethefiguretonameeachofthefollowing.a.
𝑨𝑩
𝑫
𝑪
LinePointsCollinearpointsNoncollinearpoints
PointsLinesandPlanes
SampleProblem1: Usethefiguretonameeachofthefollowing.a.
𝑨𝑩
𝑫
𝑪
Line𝑨𝑩Points𝑨,𝑩, 𝑪𝐚𝐧𝐝𝑫Collinearpoints𝑨,𝑩Noncollinearpoints𝑨, 𝑪,𝑫
PointsLinesandPlanes
SampleProblem1: Usethefiguretonameeachofthefollowing.b. 𝑲 Linesegment
Points
𝑳
PointsLinesandPlanes
SampleProblem1: Usethefiguretonameeachofthefollowing.b. 𝑲 Linesegment𝑲𝑳
Points𝑲, 𝑳
𝑳
PointsLinesandPlanes
SampleProblem1: Usethefiguretonameeachofthefollowing.c. 𝑹 Plane
RayPointsCoplanarpointsNoncoplanarpoints
𝑰
𝑺 𝑻
𝑶
PointsLinesandPlanes
SampleProblem1: Usethefiguretonameeachofthefollowing.c. 𝑹 Plane𝑺𝑻𝑶
Ray𝑰𝑹Points 𝑺, 𝑻, 𝑶, 𝑹𝐚𝐧𝐝𝑰Coplanarpoints𝑺, 𝑻, 𝑶Noncoplanarpoints𝑹, 𝑰
𝑰
𝑺 𝑻
𝑶
PointsLinesandPlanes
• Twoormoregeometricfiguresintersect,iftheyhaveoneormorepointsincommon.• Theintersection ofthefiguresisthesetofpointsthefigureshaveincommon.
PointsLinesandPlanes
• Postulate1-1 Throughanytwopointsthereisexactlyoneline.• Postulate1-2 Iftwodistinctlinesintersect,thentheyintersectinexactlyonepoint.• Postulate1-3 Iftwodistinctplanesintersect,thentheyintersectinexactlyoneline.• Postulate1-4 Throughanythreenoncollinearpointsthereisexactlyoneplane.•
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
a. 𝑫
𝑺
𝑻𝑩𝑾
𝒁𝑼
𝑸
𝝅
Nametheintersectionofline𝑸𝒁andsegment𝑾𝑼 ̅.
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
a. 𝑫
𝑺
𝑻𝑩𝑾
𝒁𝑼
𝑸
𝝅
Nametheintersectionofline𝑸𝒁andsegment𝑾𝑼 ̅.
Point𝑻
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
a. 𝑫
𝑺
𝑻𝑩𝑾
𝒁𝑼
𝑸
𝝅
Nametheintersectionofplane𝝅 andline𝑫𝑩.
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
a. 𝑫
𝑺
𝑻𝑩𝑾
𝒁𝑼
𝑸
𝝅
Nametheintersectionofplane𝝅 andline𝑫𝑩.
Point𝑺
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
a. 𝑫
𝑺
𝑻𝑩𝑾
𝒁𝑼
𝑸
𝝅
Namethetwooppositeraysatpoint𝑻.
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
a. 𝑫
𝑺
𝑻𝑩𝑾
𝒁𝑼
𝑸
𝝅
Namethetwooppositeraysatpoint𝑻.
𝑻𝑸and𝑻𝒁
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
a. 𝑫
𝑺
𝑻𝑩𝑾
𝒁𝑼
𝑸
𝝅
Whatisanothernameforplane𝝅?
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
a. 𝑫
𝑺
𝑻𝑩𝑾
𝒁𝑼
𝑸
𝝅
Whatisanothernameforplane𝝅?
Plane 𝑻𝑺𝑼
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
b.
𝑴𝑪
𝑻
𝑩
𝑮𝑳
𝑺
𝝅
Nametheintersectionofplane𝝅 andplane𝝉.
𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
b.
𝑴𝑪
𝑻
𝑩
𝑮𝑳
𝑺
𝝅
Nametheintersectionofplane𝝅 andplane𝝉.
Line𝑩𝑺
𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
b.
𝑴𝑪
𝑻
𝑩
𝑮𝑳
𝑺
𝝅
Whatisanothernameforplane𝝅?
𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
b.
𝑴𝑪
𝑻
𝑩
𝑮𝑳
𝑺
𝝅
Whatisanothernameforplane𝝅?
𝝉
Plane 𝑳𝑴𝑮
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
b.
𝑴𝑪
𝑻
𝑩
𝑮𝑳
𝑺
𝝅
Nametheintersectionofline𝑴𝑮 andline𝑩𝑺.
𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
b.
𝑴𝑪
𝑻
𝑩
𝑮𝑳
𝑺
𝝅
Nametheintersectionofline𝑴𝑮 andline𝑩𝑺.
𝝉
Point 𝑪
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
b.
𝑴𝑪
𝑻
𝑩
𝑮𝑳
𝑺
𝝅
Nameapointthatiscollinearwith𝑴and𝑪.
𝝉
Point 𝑮
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
c.
𝑪
𝑳
𝑩 𝑱
𝑯 𝑷
𝝅
Nametheintersectionofplane𝝅 andline𝑳𝑪.
𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
c.
𝑪
𝑳
𝑩 𝑱
𝑯 𝑷
𝝅
Nametheintersectionofplane𝝅 andline𝑳𝑪.
Point𝑪𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
c.
𝑪
𝑳
𝑩 𝑱
𝑯 𝑷
𝝅
Nametheintersectionofplane𝝉 andline𝑳𝑪.
𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
c.
𝑪
𝑳
𝑩 𝑱
𝑯 𝑷
𝝅
Nametheintersectionofplane𝝉 andline𝑳𝑪.
Point𝑳𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
c.
𝑪
𝑳
𝑩 𝑱
𝑯 𝑷
𝝅
Nameapointthatiscoplanarwith𝑯and𝑳.
𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
c.
𝑪
𝑳
𝑩 𝑱
𝑯 𝑷
𝝅
Nameapointthatiscoplanarwith𝑯and𝑳.
Point𝑷𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
c.
𝑪
𝑳
𝑩 𝑱
𝑯 𝑷
𝝅
Nametheoppositerayofray𝑪𝑩.
𝝉
PointsLinesandPlanes
SampleProblem2: Refertoeachfigure.
c.
𝑪
𝑳
𝑩 𝑱
𝑯 𝑷
𝝅
Nametheoppositerayofray𝑪𝑩.
Ray𝑪𝑱𝝉
PointsLinesandPlanes
SampleProblem3: Drawandlabelfigureforeachrelationship.
a. • Plane𝑨𝑩𝑺containslines𝑨𝑩,𝑪𝑫,and𝑨𝑲.• Lines𝑨𝑩 and𝑪𝑫 intersectinpoint𝑮.• Lines𝑪𝑫and𝑨𝑲 intersectinpoint𝑺.• Lines𝑨𝑩 and𝑨𝑲intersectinpoint𝑨.
PointsLinesandPlanes
SampleProblem3: Drawandlabelfigureforeachrelationship.
a.
𝑮
𝑺
𝑩
𝑲𝑫
𝑪
• Plane𝑨𝑩𝑺containslines𝑨𝑩,𝑪𝑫,and𝑨𝑲.• Lines𝑨𝑩 and𝑪𝑫 intersectinpoint𝑮.• Lines𝑪𝑫and𝑨𝑲 intersectinpoint𝑺.• Lines𝑨𝑩 and𝑨𝑲intersectinpoint𝑨.
𝑨
PointsLinesandPlanes
SampleProblem3: Drawandlabelfigureforeachrelationship.
b. • Plane𝝅 containsline𝑨𝑩andpoint𝑳.
• Plane𝝉 containsline𝑬𝑭andpoint𝑺.
• Lines𝑨𝑩and 𝑬𝑭intersectinpoint𝑯.
• Theintersectionofplane𝝅 andplane𝝉 isline 𝑳𝑼.
PointsLinesandPlanes
SampleProblem3: Drawandlabelfigureforeachrelationship.
b. • Plane𝝅 containsline𝑨𝑩andpoint𝑳.
• Plane𝝉 containsline𝑬𝑭andpoint𝑺.
• Lines𝑨𝑩and 𝑬𝑭intersectinpoint𝑯.
• Theintersectionofplane𝝅 andplane𝝉 isline 𝑳𝑼.
𝝅
𝝉
𝑯𝑨
𝑩
𝑺
𝑬
𝑭𝑳
𝑼
PointsLinesandPlanes
SampleProblem3: Drawandlabelfigureforeachrelationship.
c. • Plane𝝅 andplane𝝉 donothasintersect.• Plane𝜺 intersectplane𝝅 inline𝑩𝑪.• Plane𝜺 intersectplane𝝉 inline𝑬𝑹.
PointsLinesandPlanes
SampleProblem3: Drawandlabelfigureforeachrelationship.
c. • Plane𝝅 andplane𝝉 donothasintersect.• Plane𝜺 intersectplane𝝅 inline𝑩𝑪.• Plane𝜺 intersectplane𝝉 inline𝑬𝑹.
𝝅
𝜺𝝉
𝑩
𝑪
𝑹
𝑬