Introduction to Points, Lines, and Rays
The fundamental rules of geometry go all the way back to the time of the ancient Egyptians and Greeks, who used geometry to calculate the diameter of the earth and the distance to the moon. They employed the laws of Euclidean geometry (named after Euclid, a Greek mathematician who lived in the 3rd century ). Euclidean plane geometry involves points and lines on perfectly flat surfaces.
In plane geometry, certain starting concepts aren’t defined formally, but are considered intuitively obvious. The point and the line are examples. A point can be envisioned as an infinitely tiny sphere, having height, width, and depth all equal to zero, but nevertheless possessing a specific location. A line can be thought of as an infinitely thin, perfectly straight, infinitely long wire.
Naming Points And Lines
Points and lines are usually named using uppercase, italicized letters of the alphabet. The most common name for a point is P (for “point”), and the most common name for a line is L (for “line”). If multiple points are involved in a scenario, the letters immediately following P are used, for example Q , R , and S . If two or more lines exist in a scenario, the letters immediately following L are used, for example M and N . Alternatively, numeric subscripts can be used with P and L . Then we have points called P _{1} , P _{2} , P _{3} , and so forth, and lines called L _{1} , L _{2} , L _{3} , and so forth.
Two Point Principle
Suppose that P and Q are two different geometric points. Two distinct points define one and only one (that is, a unique) line L . The following two statements are always true, as shown in Fig. 1-1:
P and Q lie on a common line L
L is the only line on which both points lie
Fig. 1-1 . Two point principle. For two specific points P and Q , line L is unique.
Distance Notation
The distance between any two points P and Q , as measured from P towards Q along the straight line connecting them, is symbolized by writing PQ . Units of measurement such as meters, feet, millimeters, inches, miles, or kilometers are not important in pure mathematics, but they are important in physics and engineering. Sometimes a lowercase letter, such as d , is used to represent the distance between two points.
Line Segments, Rays, and the Midpoint Principle
Line Segments
The portion of a line between two different points P and Q is called a line segment . The points P and Q are called the end points . A line segment can theoretically include both of the end points, only one of them, or neither of them.
If a line segment contains both end points, it is a closed line segment . If it contains one of the end points but not the other, it is a half-open line segment . If it contains neither end point, it is an open line segment . Whether a line segment is closed, half-open, or open, its length is the same. Adding or taking away a single point makes no difference, mathematically, in the length because points have zero size in all dimensions! Yet the conceptual difference between these three types of line segments is like the difference between daylight, twilight, and darkness.
Rays (Half Lines)
Sometimes, mathematicians talk about the portion of a geometric line that lies “on one side” of a certain point. In Fig. 1-1, imagine the set of points that starts at P , then passes through Q , and extends onward past Q forever. This is known as a ray or half line.
Fig. 1-1 . Two point principle. For two specific points P and Q , line L is unique.
The ray defined by P and Q might include the end point P , in which case it is a closed-ended ray . If the end point is left out, the theoretical object is an open-ended ray . In either case, the ray is said to “begin” at point P ; informally we might say that it is either “tacked down at the end” or “dangling at the end.”
Midpoint Principle
Suppose there is a line segment connecting two points P and R . Then there is one and only one point Q on the line segment such that PQ = QR , as shown in Fig. 1-2.
Fig. 1-2 . Midpoint principle. Point Q is unique.
Points, Lines, and Rays Practice Problems
PROBLEM 1
Suppose, in Fig. 1-2, we find the midpoint Q _{2} between P and Q , then the midpoint Q _{3} between P and Q _{2} , then the midpoint Q _{4} between P and Q _{3} , and so on. In mathematical language, we say we keep finding midpoints Q _{( n +1)} between P and Q _{n} , where n is a positive whole number. How long can this process go on?
Fig. 1-2 . Midpoint principle. Point Q is unique.
SOLUTION 1
The process can continue forever. In theoretical geometry, there is no limit to the number of times a line segment can be cut in half. This is because a line segment contains an infinite number of points.
PROBLEM 2
Suppose we have a line segment with end points P and Q . What is the difference between the distance PQ and the distance QP ?
SOLUTION 2
This is an interesting question. If we consider distance without paying attention to the direction in which it is measured, then PQ = QP . But if direction is important, we define PQ = − QP .
In basic plane geometry, direction is sometimes specified in diagrams in order to get viewers to move their eyes from right to left instead of from left to right, or from bottom to top rather than from top to bottom.
Practice problems of these concepts can be found at: Geometry Basic Rules Practice Test.
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From Geometry Demystified: A Self-Teaching Guide. Copyright © 2003 by The McGraw-Hill Companies, Inc. All Rights Reserved.