1.2 Points, Lines & Planes

7/24/2019 1.2 Points, Lines & Planes

1/25

1.2 Points, Lines and1.2 Points, Lines and

PlanesPlanesGeometryGeometry

Mrs. SpitzMrs. SpitzFall 2005Fall 2005


2/25

Objecties!"ssi#nment$Objecties!"ssi#nment$

%nderstand and &se t'e basic%nderstand and &se t'e basic

&nde(ned terms and de(ned terms&nde(ned terms and de(ned terms

o) #eometry.o) #eometry.

S*etc' t'e intersections o) lines andS*etc' t'e intersections o) lines and

planes.planes.

"ssi#nment$ pp. 1+15 -12 all"ssi#nment$ pp. 1+15 -12 all


3/25

%sin# %nde(ned terms and%sin# %nde(ned terms and

de(nitionde(nition" defnition&ses *no/n /ords to

describe a ne/ /ord. n #eometry,some /ords s&c' as point, line andplane are undefnedterms or not)ormally de(ned.


4/25

ample /'at yo&r note pa#eample /'at yo&r note pa#e

s'o&ld loo* li*e no/s'o&ld loo* li*e no/

Name____________________________

Date___________________Period_____

Class_____________________________

1.2 Points,

Lines, &

Planes

Objectives:

Assignment:sing

!nde"ined

terms and

de"initions

1.1. nderstand and !se t#e basic !nde"inednderstand and !se t#e basic !nde"ined

terms and de"ined terms o" geometr$.terms and de"ined terms o" geometr$.

2.2. %etc# t#e intersections o" lines and%etc# t#e intersections o" lines and

'lanes.'lanes.

Assignment: ''. 1()1* +1)2 allAssignment: ''. 1()1* +1)2 all

A de"inition !ses no-n -ords to describe a

ne- -ord. n geometr$, some -ords s!c#

as 'oint, line and 'lane are !nde"ined

terms or not "ormall$ de"ined.


5/25


de(nitionde(nition

3 "point'as nodimension. t is&s&allyrepresented by asmall dot.

A

Point "


6/25


de(nitionde(nition3 " lineetends in one

dimension. t is&s&ally representedby a strai#'t line /it't/o arro/'eads toindicate t'at t'e lineetends /it'o&t endin t/o directions. n

t'is boo*, lines areal/ays strai#'t lines.

A

B

l

Line lor "4


7/25


de(nitionde(nition"planeetends in t/o

dimensions. t is&s&ally represented bya s'ape t'at loo*s li*e

a tabletop or /all. o&m&st ima#ine t'at t'eplane etends /it'o&tend een t'o' t'e

dra/in# o) a planeappears to 'aeed#es.

A

B

C

M

Plane M or plane"46


8/25

" )e/ basic concepts . . ." )e/ basic concepts . . .

M&st be commonly &nderstoodM&st be commonly &nderstood

/it'o&t bein# de(ned. One s&c'/it'o&t bein# de(ned. One s&c'

concept is t'e idea t'at a point liesconcept is t'e idea t'at a point lies

on a line or a plane.on a line or a plane.

Collinear pointsCollinearpointsare points t'at lie onare points t'at lie on

t'e same line.t'e same line.

Coplanar pointsCoplanar pointsare points t'at lie onare points t'at lie on

t'e same plane.t'e same plane.


9/25

. 1$ 7amin# 6ollinear and. 1$ 7amin# 6ollinear and

6oplanar Points6oplanar Pointsa.a. 7ame t'ree7ame t'ree

points t'at arepoints t'at are

collinearcollinear

Sol&tion$Sol&tion$

8, and F lie on t'e8, and F lie on t'e

same line, sosame line, sot'ey are collinear.t'ey are collinear.

G

D E F

H


10/25


6oplanar Points6oplanar Pointsb.b. 7ame )o&r points7ame )o&r points

t'at are coplanar.t'at are coplanar.

Sol&tion$Sol&tion$8, , F, and G lie on t'e8, , F, and G lie on t'e

same plane, so t'eysame plane, so t'ey

are coplanar. "lso 8,are coplanar. "lso 8,

, F, and 9 are, F, and 9 are

coplanar: alt'o',coplanar: alt'o',

t'e plane containin#t'e plane containin#

t'em is not dra/n.t'em is not dra/n.

G

D E F

H


11/25


6oplanar Points6oplanar Pointsc.c. 7ame t'ree points7ame t'ree points

t'at are nott'at are notcollinear.collinear.

Sol&tion$Sol&tion$

;'ere are many;'ere are manycorrect ans/ers.correct ans/ers.

For instance,For instance,points 9, , and Gpoints 9, , and Gdo not lie on t'edo not lie on t'esame line.same line.

G

D E F

H


12/25

More . . .More . . .

"not'er &nde(ned"not'er &nde(ned

concept in #eometry isconcept in #eometry is

t'e idea t'at a pointt'e idea t'at a point

on a line is bet/eenon a line is bet/een

t/o ot'er points ont/o ot'er points ont'e line. o& can &set'e line. o& can &se

t'is idea to de(net'is idea to de(ne

ot'er important termsot'er important terms

in #eometry.in #eometry. 6onsider t'e line "46onsider t'e line "4


13/25


14/25


;'e ray "4;'e ray "4


15/25


7ote t'at "4 is t'e7ote t'at "4 is t'e

same as 4" and "4same as 4" and "4

is t'e same as 4".is t'e same as 4".

9o/eer, "4 and9o/eer, "4 and4" are not t'e4" are not t'e

same. ;'ey 'aesame. ;'ey 'ae

di?erent initialdi?erent initial

points and etendpoints and etendin di?erentin di?erent

directions.directions.

l

Line lor "4

A

A

B

B

>ay 4"


16/25


) 6 is bet/een " and) 6 is bet/een " and4, t'en 6" and 64 are4, t'en 6" and 64 are

opposite rays.opposite rays.

Li*e points, se#mentsLi*e points, se#ments

and rays are collinearand rays are collinear

i) t'ey lie on t'e samei) t'ey lie on t'e same

line. So, any t/oline. So, any t/o

opposite rays areopposite rays arecollinear. Se#ments,collinear. Se#ments,

rays and lines arerays and lines are

coplanar i) t'ey lie oncoplanar i) t'ey lie on

t'e same plane.t'e same plane.

l

Line lor "4

A

C

B


17/25

. 2$ 8ra/in# lines, se#ments. 2$ 8ra/in# lines, se#ments

and raysand rays 8ra/ t'ree noncollinear points @, A,8ra/ t'ree noncollinear points @, A,

and L. ;'en dra/ @A, AL and [email protected] L. ;'en dra/ @A, AL and L@.

J

K

L

8ra/ @, A and L

;'en dra/ @A


18/25




J

K

L

8ra/ AL


19/25




J

K

L

8ra/ L@


20/25

. +$ 8ra/in# Opposite. +$ 8ra/in# Opposite

>ays>ays 8ra/ t/o lines. Label8ra/ t/o lines. Labelpoints on t'e lines andpoints on t'e lines and

name t/o pairs o)name t/o pairs o)


Sol&tion$ Points M, 7,Sol&tion$ Points M, 7,

and B are collinearand B are collinear

and B is bet/een Mand B is bet/een M

and 7. So BM and B7and 7. So BM and B7

are opposite rays.are opposite rays.

P

MQ

N

X


21/25

. +$ 8ra/in# Opposite. +$ 8ra/in# Opposite

>ays>ays 8ra/ t/o lines. Label8ra/ t/o lines. Labelpoints on t'e lines andpoints on t'e lines and

name t/o pairs o)name t/o pairs o)


Sol&tion$ Points P, C,Sol&tion$ Points P, C,

and B are collinearand B are collinear

and B is bet/een Pand B is bet/een P

and C. So BP and BCand C. So BP and BC

are opposite rays.are opposite rays.

P

MQ

N

X


22/25

Goal 2$ S*etc'in#Goal 2$ S*etc'in#

ntersections o) Lines andntersections o) Lines and

PlanesPlanes;/o or more #eometric intersect i) t'ey;/o or more #eometric intersect i) t'ey'ae one or more points in common.'ae one or more points in common.

;'e intersection o) t'e (#&res is t'e set;'e intersection o) t'e (#&res is t'e set

o) points t'e (#&res 'ae in common.o) points t'e (#&res 'ae in common. "ctiity$ p. 12 D Modelin# intersections."ctiity$ p. 12 D Modelin# intersections.

D %se t/o inde cards. Label t'em as s'o/n%se t/o inde cards. Label t'em as s'o/n

and c&t slots alon# eac' card.and c&t slots alon# eac' card. 6omplete t'e eercise and place t'e completed6omplete t'e eercise and place t'e completed

E&estions in yo&r lab section labelin# t'is LabE&estions in yo&r lab section labelin# t'is Lab

1.2.1.2.


23/25

. $ S*etc'in#. $ S*etc'in#

intersectionsintersections S*etc' t'e (#&reS*etc' t'e (#&redescribed.described.

a.a. " line t'at intersects" line t'at intersects

a plane in one pointa plane in one point 8ra/ a plane and a8ra/ a plane and aline.line.

mp'asize t'e pointmp'asize t'e point

/'ere t'ey meet./'ere t'ey meet. 8as'es indicate8as'es indicate

/'ere t'e line is/'ere t'e line is'idden by t'e plane'idden by t'e plane


24/25

. $ S*etc'in#. $ S*etc'in#

intersectionsintersections S*etc' t'e (#&reS*etc' t'e (#&redescribed.described.

b.b. ;/o planes t'at;/o planes t'at

intersect in a lineintersect in a line

8ra/ t/o planes.8ra/ t/o planes.

mp'asize t'e linemp'asize t'e line

/'ere t'ey meet./'ere t'ey meet.

8as'es indicate8as'es indicate/'ere one plane is/'ere one plane is

'idden by t'e ot'er'idden by t'e ot'er

plane.plane.


25/25

6omin# &p soon . . .6omin# &p soon . . .

C&iz a)ter 1.+C&iz a)ter 1.+

Practice E&iz on pa#e 25Practice E&iz on pa#e 25

Documents

1.2 Points, Lines & Planes