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Point, Line, Plane

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Point, Line, Plane

- Undefined terms: words that do not have a formal definition but there is agreement about what they mean.
- Defined terms: Terms that can be described using known words
- Postulate or Axiom: Rule that is accepted without proof.
- Theorem: Rule that can be proved.

Point, Line, Plane

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

A

C

Point, Line, Plane

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 -

n

A

B

C

Point, Line, Plane

Collinear Points

- Collinear points are points that lie on the same line. (The line does not have to be visible.)
- A point lies on the line if the coordinates of the point satisfy the equation of the line.
Ex: To find if A (1, 0) is collinear with

the points on the line y = -3x + 3.

Substitute x = 1 and y = 0 in the equation.

0 = -3 (1) + 3

0 = -3 + 3

0 = 0

The point A satisfies the equation, therefore the point is collinear

with the points on the line.

A

B

C

Collinear

C

A

B

Non collinear

Lesson 1-1 Point, Line, Plane

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

M

B

C

Horizontal Plane

Vertical Plane

Other

Lesson 1-1 Point, Line, Plane

Different planes in a figure:

A

B

Plane ABCD

Plane EFGH

Plane BCGF

Plane ADHE

Plane ABFE

Plane CDHG

Etc.

D

C

E

F

H

G

Lesson 1-1 Point, Line, Plane

Other planes in the same figure:

Any three non collinear points determine a plane!

Plane AFGD

Plane ACGE

Plane ACH

Plane AGF

Plane BDG

Etc.

Lesson 1-1 Point, Line, Plane

Coplanar Objects

Coplanar objects (points, lines, etc.) are objects that lie on the same plane. The plane does not have to be visible.

Are the following points coplanar?

A, B, C ?

Yes

A, B, C, F ?

No

H, G, F, E ?

Yes

E, H, C, B ?

Yes

A, G, F ?

Yes

C, B, F, H ?

No

Lesson 1-1 Point, Line, Plane

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

Line m and line n intersect at point P.

P

n

Continued…….

Lesson 1-1 Point, Line, Plane

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

Intersection of Two Planes is a Line.

B

P

A

R

Plane P and Plane R intersect at the line

Lesson 1-1 Point, Line, Plane

RA : RA and all points Y such that

A is between R and Y.

( the symbol RA is read as “ray RA” )

RayDefinition:

How to sketch:

How to name:

Lesson 1-2: Segments and Rays

Opposite Rays

Definition:

If A is between X and Y, AX and AY are opposite rays.

( Opposite rays must have the same “endpoint” )

opposite rays

not opposite rays

Lesson 1-2: Segments and Rays

Segment

Part of a line that consists of two points called the endpoints and all points between them.

Definition:

How to sketch:

How to name:

AB (without a symbol) means the length of the segment or the distance between points A and B.

Lesson 1-2: Segments and Rays

AC + CB = AB

x + 2x = 12

3x = 12

x = 4

The Segment Addition PostulatePostulate:

If C is between A and B, then AC + CB = AB.

If AC = x , CB = 2x and AB = 12, then, find x, AC and CB.

Example:

2x

x

Step 1: Draw a figure

Step 2: Label fig. with given info.

Step 3: Write an equation

x = 4

AC = 4

CB = 8

Step 4: Solve and find all the answers

Lesson 1-2: Segments and Rays