Geometry and Trigonometry
Basics of Geometry
What are some of the basic “building blocks” of geometry?
There are several basic “building blocks” of geometry, all of which have to do with the objects we often see in geometry. A zero-dimensional object that is specifically located in n-dimensional space using n coordinates is called a point. The idea of a point may be obvious to most people, but for mathematicians, describing and dealing with points is not straightforward. For example, Euclid once gave a vague definition of a point as “that which has no part.”
Euclid also called the line a “breadthless length,” and further called a straight line one that “lies evenly with the points on itself.” Modern mathematicians define lines as one-dimensional objects (although they may be part of a higher-dimensional space). They are mathematically defined as a theoretical course of a moving point that is thought to have length but no other dimension. They are often called straight lines, or by the archaic term, a right line, to emphasize the fact that there are no curves anywhere along the entire length. It is interesting to note that when geometry is used in an axiomatic system, a line is usually considered an undefined term (for more about axiomatic systems, see “Foundations of Mathematics”). In analytic geometry, a line is defined by the basic equation ax + by = c, in which a, b, and c are any number, but a and b can’t be zero at the same time.
A line segment is the shortest curve—which is actually straight—to connect two different points. It is a finite portion of an infinite line. Line segments are most often labeled with two letters corresponding to the endpoints of the line segment, such as a and b, and written as ab.
Distance is the length of the path between two points, or the length of a segment, for example, between points a and b. When talking about the distance between any two points associated with a and b (as real numbers) on a number line, distance becomes the absolute value of b - a (|b - a|).
Another basic building block is the ray. Think of a ray as a laser beam: It originates at one point and continues in one direction toward infinity. A ray is defined as part of a line on one side of a point, and includes that point. Two letters are needed to name a ray. For example, Ray AB is defined by point A, where it begins, and point B, the point the ray goes through. (But note: Ray AB is not the same as ray BA.) If the initial point is not included, the resulting figure is called a half line. In geometry, a ray is usually considered a half-infinite line with one of the two points A and B taken to be at infinity.