Sec. 1.1 – Points, Lines, Planes
Aug. 26-27, 2010
Everything we see around us has some basis in Geometry, so it should
come as no surprise that the word literally means “to measure the earth.”
More specifically, Geometry involves the study of solids, surfaces, points,
lines, curves, angles and their relationships in space.
Some people think of Geometry as a type of game. Like many other
games or systems, we need a place to start. We need some game pieces
to move around the board and some basic rules for playing the game.
In our game of Geometry, the basic game pieces are the undefined terms
of Geometry and the basic rules are called postulates.
The Undefined Terms of Geometry
Point is one of three basic undefined terms in geometry.
- A point has no size and no dimension. It merely determines a location or
position.
- A point is generally represented in a drawing by a dot and named with a
single capital letter. We do not have to use any type of symbol when
naming a point, though some people will place the letters “pt” in front of
the capital letter for clarification.
Line is also an undefined term in geometry.
- A line consists of infinitely many points extending without end in opposite
directions.
- A line has length, but no width. Thus a line has only one dimension.
- On a line, we measure distance.
- A line is usually named using any two points that lie on the line or with a
single lower case letter. We place a double-ended arrow above the two
letters to indicate a line.
Plane is the third undefined term of geometry.
- A plane can be thought of as a flat surface that extends indefinitely in all
directions.
- Although a plane has no boundaries, it is usually represented in diagrams
by a four-sided figure (parallelogram).
- A plane has two dimensions, called length and width, but no thickness.
- On a plane, we measure area.
- A plane is named by a single capital letter written in script or cursive or by
any three points that lie in the plane ( an do not lie on the same line).
When naming a plane, we place the word “plane” in front of the script
letter or the three point to indicate the specific set of points.
Definitions in Geometry
A definition is a statement that clarifies or explains the meaning of a word
or phrase. It is impossible, however, to “define” point, line, or plane without
using words or phrases that themselves need definition. So these terms
remain undefined.
Yet these terms serve as the basis for all of geometry and we can use them
to develop definitions for many of the terms we discuss this year.
For example in geometry we define space as the set of all points.
- Space has three dimensions - length, width and height.
- In space, we measure volume.
# of
dimensions
None
Name of
figure
Point
Dimensions
called …
----------
On this figure, we
measure …
----------
One
Line
Length
Distance
Two
Plane
Area
Three
Space
Length, Width
Length, Width,
and Depth
Volume
A segment is a subset (part) of a line consisting of two points, called
endpoints, and all points on the line that lie between those two points.
The symbol used for identifying a segment is a bar placed over the letters
assigned to its two endpoints. For example, AB denotes the segment with
endpoints at points A and B. When naming a segment, you can use the
endpoints in either order. Thus (segment) AB and (segment) BA name the
same set of points.
The symbol used to denote a line is similar, but the naming process is
slightly different. The symbol used for identifying a line is a double-ended
arrow placed above the letters assigned to ANY two points which lie on the
line. For example, AB denotes a line which contains points A and B. Like
a segment, (line) AB and (line) BA name the same set of points.
A second subset of a line that we will discuss is a ray. Like a segment, a
ray has a starting point, called its endpoint. Unlike a segment, a ray does
not have a second endpoint – it continues indefinitely in one direction. As a
result, a ray is also know as a “half-line.”
Formally, a ray is a subset of a line that contains (segment) AB and all
points C that lie on (line) AB such that point B is between points A and C.
In this case, A would be the endpoint and B and C would be points that lie
on the ray.
Unlike naming procedures for a segment and a line, the order in which you
select points for naming a ray is very important. The first point is the
endpoint (tells us where to start) and the second point is any other point that
lies on the ray (gives direction).
Thus (ray) AB names a ray that starts at point A and goes in the direction of
point B and (ray) BA names a ray that starts at point B and goes in the
direction of point A. These two rays are NOT the same – they actually
travel in opposite directions on (line) AB.
However, if point B lies between points A and C, then (ray) AB and
(ray) AC wold name the same set of points and thus the same ray.