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