lesson 1-2 point, line, plane 1 lesson 1-2 point, line, plane
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Lesson 1-2 Point, Line, Plane 1
Lesson 1-2
Point, Line, Plane
Objectives/Assignment:
Understand and use the basic undefined terms and defined terms of geometry.
Sketch the intersections of lines and planes.
Using Undefined terms and definition
A definition uses known words to describe a new word. In geometry, some words such as point, line and plane are undefined terms or not formally defined.
Using Undefined terms and definition
• A point has no dimension. It is usually represented by a small dot.
A
Point A
Lesson 1-1 Point, Line, Plane 5
Points Points do not have actual size.
How to Sketch:
Using dots
How to label:
Use capital letters
Never name two points with the same letter (in the same sketch).
A
B AC
Lesson 1-1 Point, Line, Plane 6
Lines Lines extend indefinitely and have no thickness or width. How to sketch : using arrows at both ends.
How to name: 2 ways(1) small script letter – line n(2) any two points on the line -
Never name a line using three points - , , , , ,AB BC AC BA CA CB
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nA
BC
ABC�������������� �
Using Undefined terms and definitionA plane extends in two
dimensions. It is usually represented by a shape that looks like a tabletop or wall. You must imagine that the plane extends without end even though the drawing of a plane appears to have edges.
A
BC
M
Plane M or plane ABC
Lesson 1-1 Point, Line, Plane 8
Planes
A plane is a flat surface that extends indefinitely in all directions. How to sketch: Use a parallelogram (four sided figure) How to name: 2 ways
(1) Capital script letter – Plane M(2) Any 3 non collinear points in the plane - Plane: ABC/ ACB / BAC /
BCA / CAB / CBA
A
BC
Horizontal Plane
M
Vertical Plane Other
A few basic concepts . . .
Must be commonly understood without being defined. One such concept is the idea that a point lies on a line or a plane.
Collinear points are points that lie on the same line.
Coplanar points are points that lie on the same plane.
Ex. 1: Naming Collinear and Coplanar Points
a. Name three points that are collinear
Solution:
D, E and F lie on the same line, so they are collinear.
G
D E F
H
Ex. 1: Naming Collinear and Coplanar Pointsb. Name four points that
are coplanar.
Solution:
D, E, F, and G lie on the same plane, so they are coplanar. Also D, E, F, and H are coplanar; although, the plane containing them is not drawn.
G
D E F
H
Ex. 1: Naming Collinear and Coplanar Pointsc. Name three points
that are not collinear.
Solution: There are many correct
answers. For instance, points H, E, and G do not lie on the same line.
G
D E F
H
Lesson 1-1 Point, Line, Plane 13
Different planes in a figure:A B
CD
EF
GH
Plane ABCD
Plane EFGH
Plane BCGF
Plane ADHE
Plane ABFE
Plane CDHG
Etc.
Lesson 1-1 Point, Line, Plane 14
Other planes in the same figure:
Any three non collinear points determine a plane!
H
E
G
DC
BA
F
Plane AFGD
Plane ACGE
Plane ACH
Plane AGF
Plane BDG
Etc.
Lesson 1-1 Point, Line, Plane 15
Coplanar Objects
Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.
H
E
G
DC
BA
F
Are the following points coplanar?
A, B, C ?A, B, C, F ?H, G, F, E ?E, H, C, B ?A, G, F ?C, B, F, H ?
YesNo
YesYesYesNo
More . . .
Another undefined concept in geometry is the idea that a point on a line is between two other points on the line. You can use this idea to define other important terms in geometry.
Consider the line AB (symbolized by AB).
l
Line l or AB
More . . .
The line segment or segment AB (symbolized by AB) consists of the endpoints A and B, and all points on AB that are between A and B.
l
Line l or AB
A
A
B
B
Segment AB
More . . .
The ray AB (symbolized by AB) consists of the initial point A and all points on AB that lie on the same side of A as point B.
l
Line l or AB
A
A
B
BRay AB
More . . .
Note that AB is the same as BA and AB is the same as BA. However, AB and BA are not the same. They have different initial points and extend in different directions.
l
Line l or AB
A
A
B
B
Ray BA
More . . . If C is between A and B,
then CA and CB are opposite rays.
Like points, segments and rays are collinear if they lie on the same line. So, any two opposite rays are collinear. Segments, rays and lines are coplanar if they lie on the same plane.
l
Line l or AB
A
C
B
Ex. 2: Drawing lines, segments and rays
Draw three noncollinear points J, K, and
L. Then draw JK, KL and LJ.
J
K
L
Draw J, K and L
Then draw JK
Ex. 2: Drawing lines, segments and rays
Draw three noncollinear points J, K, and
L. Then draw JK, KL and LJ.
J
K
L
Draw KL
Ex. 2: Drawing lines, segments and rays
Draw three noncollinear points J, K, and
L. Then draw JK, KL and LJ.
J
K
L
Draw LJ
Ex. 3: Drawing Opposite Rays
Draw two lines. Label points on the lines and name two pairs of opposite rays.
Solution: Points M, N, and
X are collinear and X is
between M and N. So
XM and XN are opposite
rays.
P
MQ
N
X
Ex. 3: Drawing Opposite Rays
Draw two lines. Label points on the lines and name two pairs of opposite rays.
Solution: Points P, Q, and
X are collinear and X is
between P and Q. So XP
and XQ are opposite
rays.
P
MQ
N
X
Goal 2: Sketching Intersections of Lines and Planes Two or more geometric intersect if they
have one or more points in common. The intersection of the figures is the set of points the figures have in common.
Activity: p. 12 – Modeling intersections.• Use two index cards. Label them as shown
and cut slots along each card. • Complete the exercise and place the completed
questions in your lab section labeling this Lab 1.2.
Ex. 4: Sketching intersections Sketch the figure
described.a. A line that intersects a
plane in one point Draw a plane and a
line. Emphasize the point
where they meet. Dashes indicate
where the line is hidden by the plane
Ex. 4: Sketching intersections
Sketch the figure described.
b. Two planes that intersect in a line
Draw two planes. Emphasize the line
where they meet. Dashes indicate where
one plane is hidden by the other plane.
Lesson 1-1 Point, Line, Plane 29
Intersection of Figures
The intersection of two figures is the set of points that are common in both figures.
The intersection of two lines is a point.
m
n
P
Continued…….
Line m and line n intersect at point P.
Lesson 1-1 Point, Line, Plane 30
3 Possibilities of Intersection of a Line and a Plane
(1) Line passes through plane – intersection is a point.
(2) Line lies on the plane - intersection is a line.
(3) Line is parallel to the plane - no common points.
Lesson 1-1 Point, Line, Plane 31
Intersection of Two Planes is a Line.
P
R
A
B
Plane P and Plane R intersect at the line AB�������������� �