Points Lines and Planes - GeometryCoach.com...2015/08/01 · Points Lines and Planes •In...

PointsLinesandPlanes

Unit1Lesson2

Studentswillbeableto:

•Drawpoints,lines,linesegments,rays,andplanes.• Identifypoints,lines,linesegments,raysand

planes.Knowprecisedefinitionsofline,andlinesegment,based

ontheundefinednotionsofpoint,line.

KeyVocabulary:Aline,Apoint,

Alinesegment,Aplane,

Intersection.

• Ingeometry,somewords,suchaspoint,line,andplane,areundefinedterms.Althoughthesewordsarenotformallydefined,itisimportanttohavegeneralagreementaboutwhateachwordmeans.

• Apointhasnodimension.• Itisusuallyrepresentedbyasmalldotandnamedbyacapitalletter.

• Aline extendsinonedimension.• Itisusuallyrepresentedbyastraightlinewithtwoarrowheadstoindicatethatthelineextendswithoutendintwodirections,andisnamedbytwopointsonthelineoralowercasescriptletter.

• Aplane extendsintwodimensions.• Itisusuallyrepresentedbyashapethatlookslikeatabletoporwall.• Youmustimaginethattheplaneextendswithoutend,eventhoughthedrawingofaplaneappearstohaveedges,andisnamedbyacapitalscriptletteror3non-collinearpoints.

• Alinesegment isasetofpointsandhasaspecificlengthi.e.itdoesnotextendindefinitely.• Ithasnothicknessorwidth,isusuallyrepresentedbyastraightlinewithnoarrowheadstoindicatethatithasafixedlength,andisnamedbytwopointsonthelinesegmentwithalinesegmentsymbolabovetheletters.

• Aray isasetofpointsandextendsinonedimensioninonedirection(notintwodirections).Ithasnothicknessorwidth,isusuallyrepresentedbyastraightlinewithonearrowheadtoindicatethatitextendswithoutendinthedirectionofthearrowhead,andisnamedbytwopointsontheraywitharaysymbolabovetheletters

• Collinearpoints arepointsthatlieonthesameline.

• Coplanarpoints arepointsthatlieonthesameplane.

SampleProblem1: Usethefiguretonameeachofthefollowing.a.

𝑨𝑩

LinePointsCollinearpointsNoncollinearpoints

SampleProblem1: Usethefiguretonameeachofthefollowing.a.

𝑨𝑩

Line𝑨𝑩Points𝑨,𝑩, 𝑪𝐚𝐧𝐝𝑫Collinearpoints𝑨,𝑩Noncollinearpoints𝑨, 𝑪,𝑫

SampleProblem1: Usethefiguretonameeachofthefollowing.b. 𝑲 Linesegment

Points

SampleProblem1: Usethefiguretonameeachofthefollowing.b. 𝑲 Linesegment𝑲𝑳

Points𝑲, 𝑳

SampleProblem1: Usethefiguretonameeachofthefollowing.c. 𝑹 Plane

RayPointsCoplanarpointsNoncoplanarpoints

𝑺 𝑻

SampleProblem1: Usethefiguretonameeachofthefollowing.c. 𝑹 Plane𝑺𝑻𝑶

Ray𝑰𝑹Points 𝑺, 𝑻, 𝑶, 𝑹𝐚𝐧𝐝𝑰Coplanarpoints𝑺, 𝑻, 𝑶Noncoplanarpoints𝑹, 𝑰

𝑺 𝑻

• Twoormoregeometricfiguresintersect,iftheyhaveoneormorepointsincommon.• Theintersection ofthefiguresisthesetofpointsthefigureshaveincommon.

• Postulate1-1 Throughanytwopointsthereisexactlyoneline.• Postulate1-2 Iftwodistinctlinesintersect,thentheyintersectinexactlyonepoint.• Postulate1-3 Iftwodistinctplanesintersect,thentheyintersectinexactlyoneline.• Postulate1-4 Throughanythreenoncollinearpointsthereisexactlyoneplane.•

SampleProblem2: Refertoeachfigure.

a. 𝑫

𝑻𝑩𝑾

𝒁𝑼

Nametheintersectionofline𝑸𝒁andsegment𝑾𝑼 ̅.

a. 𝑫

𝑻𝑩𝑾

𝒁𝑼

Nametheintersectionofline𝑸𝒁andsegment𝑾𝑼 ̅.

Point𝑻

a. 𝑫

𝑻𝑩𝑾

𝒁𝑼

Nametheintersectionofplane𝝅 andline𝑫𝑩.

a. 𝑫

𝑻𝑩𝑾

𝒁𝑼

Nametheintersectionofplane𝝅 andline𝑫𝑩.

Point𝑺

a. 𝑫

𝑻𝑩𝑾

𝒁𝑼

Namethetwooppositeraysatpoint𝑻.

a. 𝑫

𝑻𝑩𝑾

𝒁𝑼

Namethetwooppositeraysatpoint𝑻.

𝑻𝑸and𝑻𝒁

a. 𝑫

𝑻𝑩𝑾

𝒁𝑼

Whatisanothernameforplane𝝅?

a. 𝑫

𝑻𝑩𝑾

𝒁𝑼

Plane 𝑻𝑺𝑼

𝑴𝑪

𝑮𝑳

Nametheintersectionofplane𝝅 andplane𝝉.

𝑴𝑪

𝑮𝑳

Nametheintersectionofplane𝝅 andplane𝝉.

Line𝑩𝑺

𝑴𝑪

𝑮𝑳

𝑴𝑪

𝑮𝑳

Plane 𝑳𝑴𝑮

𝑴𝑪

𝑮𝑳

Nametheintersectionofline𝑴𝑮 andline𝑩𝑺.

𝑴𝑪

𝑮𝑳

Nametheintersectionofline𝑴𝑮 andline𝑩𝑺.

Point 𝑪

𝑴𝑪

𝑮𝑳

Nameapointthatiscollinearwith𝑴and𝑪.

Point 𝑮

𝑩 𝑱

𝑯 𝑷

Nametheintersectionofplane𝝅 andline𝑳𝑪.

𝑩 𝑱

𝑯 𝑷

Nametheintersectionofplane𝝅 andline𝑳𝑪.

Point𝑪𝝉

𝑩 𝑱

𝑯 𝑷

Nametheintersectionofplane𝝉 andline𝑳𝑪.

𝑩 𝑱

𝑯 𝑷

Nametheintersectionofplane𝝉 andline𝑳𝑪.

Point𝑳𝝉

𝑩 𝑱

𝑯 𝑷

Nameapointthatiscoplanarwith𝑯and𝑳.

𝑩 𝑱

𝑯 𝑷

Nameapointthatiscoplanarwith𝑯and𝑳.

Point𝑷𝝉

𝑩 𝑱

𝑯 𝑷

Nametheoppositerayofray𝑪𝑩.

𝑩 𝑱

𝑯 𝑷

Nametheoppositerayofray𝑪𝑩.

Ray𝑪𝑱𝝉

SampleProblem3: Drawandlabelfigureforeachrelationship.

a. • Plane𝑨𝑩𝑺containslines𝑨𝑩,𝑪𝑫,and𝑨𝑲.• Lines𝑨𝑩 and𝑪𝑫 intersectinpoint𝑮.• Lines𝑪𝑫and𝑨𝑲 intersectinpoint𝑺.• Lines𝑨𝑩 and𝑨𝑲intersectinpoint𝑨.

𝑲𝑫

• Plane𝑨𝑩𝑺containslines𝑨𝑩,𝑪𝑫,and𝑨𝑲.• Lines𝑨𝑩 and𝑪𝑫 intersectinpoint𝑮.• Lines𝑪𝑫and𝑨𝑲 intersectinpoint𝑺.• Lines𝑨𝑩 and𝑨𝑲intersectinpoint𝑨.

b. • Plane𝝅 containsline𝑨𝑩andpoint𝑳.

• Plane𝝉 containsline𝑬𝑭andpoint𝑺.

• Lines𝑨𝑩and 𝑬𝑭intersectinpoint𝑯.

• Theintersectionofplane𝝅 andplane𝝉 isline 𝑳𝑼.

b. • Plane𝝅 containsline𝑨𝑩andpoint𝑳.

• Plane𝝉 containsline𝑬𝑭andpoint𝑺.

• Lines𝑨𝑩and 𝑬𝑭intersectinpoint𝑯.

• Theintersectionofplane𝝅 andplane𝝉 isline 𝑳𝑼.

𝑯𝑨

𝑭𝑳

c. • Plane𝝅 andplane𝝉 donothasintersect.• Plane𝜺 intersectplane𝝅 inline𝑩𝑪.• Plane𝜺 intersectplane𝝉 inline𝑬𝑹.

𝜺𝝉

Points Lines and Planes - GeometryCoach.com...2015/08/01 · Points Lines and Planes •In...

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