## 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.