Geometry/Points, Lines, Line Segments and Rays

1.1: Points, Lines, Line Segments, and Rays

Points and lines are two of the most fundamental concepts in Geometry, but they are also the most difficult to define. We can describe intuitively their characteristics, but there is no set definition for them: they, along with the plane, are the undefined terms of geometry. All other geometric definitions and concepts are built on the undefined ideas of the point, line and plane. Nevertheless, we shall try to define them.

Point
A point is an exact location in space. A point is denoted by a dot. A point has no size.

Line
As for a line segment, we specify a line with two endpoints. Starting with the corresponding line segment, we find other line segments that share at least two points with the original line segment. In this way we extend the original line segment indefinitely. The set of all possible line segments findable in this way constitutes a line. A line extends indefinitely in a single dimension. Its length, having no limit, is infinite. Like the line segments that constitute it, it has no width or height. You may specify a line by specifying any two points within the line. For any two points, only one line passes through both points. On the other hand, an unlimited number of lines pass through any single point.

Ray
We construct a ray similarly to the way we constructed a line, but we extend the line segment beyond only one of the original two points. A ray extends indefinitely in one direction, but ends at a single point in the other direction. That point is called the end-point of the ray. Note that a line segment has two end-points, a ray one, and a line none. An angle can be formed when two rays meet at a common point. The rays are the sides of the angle. The point of the end of two rays is called the vertex.

Plane
A point exists in zero dimensions. A line exists in one dimension, and we specify a line with two points. A plane exists in two dimensions. We specify a plane with three points. Any two of the points specify a line. All possible lines that pass through the third point and any point in the line make up a plane. In more obvious language, a plane is a flat surface that extends indefinitely in its two dimensions, length and width. A plane has no height.

Space
Space exists in three dimensions. Space is made up of all possible planes, lines, and points. It extends indefinitely in all directions.

N-dimensional Space
Mathematics can extend space beyond the three dimensions of length, width, and height. We then refer to "normal" space as 3-dimensional space. A 4-dimensional space consists of an infinite number of 3-dimensional spaces. Etc.

Conceptual Questions
 Problem 1.1 (Geometry in Real Life) Give the geometric term(s) that is best modeled by each.

a. The location of San Francisco, California b. The surface of a chalkboard c. The tip of a pencil d. A piano chord e. The edge of a desk f. A knot in a rope g. A telephone pole h. Two connected walls i. A partially opened folder

 Problem 1.2 (Name the Plane) Use the appropriate notation to name the following plane in two different ways.

 Problem 1.3 (Name the Line) Use the appropriate notation to name the following line in five different ways.