Points, Lines, and Rays Help (page 2)

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.