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Building Blocks of Geometry

Lines

Problems

Problems

Lines

A line is the infinite set of all points arrayed in a straight formation. It is difficult to formally define, but easy to understand. A line has no thickness, it has only length, and can be named by any two points on that line. For example, a line can be called line AB, or symbolized:

Figure %: The symbol for line AB
A line can also be given a single letter as a name, such as p , and be called line p .

To form a line, take any two points, A and B, and draw a straight line through them. The line AB looks like this on paper:

Figure %: Line AB

A line extends in both directions without bound; this is why lines are usually depicted with arrows on each end. Its length is infinite, and between any two points on a line, there lie an infinite number of other points. Do you see why? You can choose two points on a line that seem to lie very close to each other, but if you "zoom in" on these points, you can always identify a point halfway between them. Then you can repeat the process with one of the original "close" points and the new halfway point to identify another point in between the two "close" points. This way you can find an infinite number of points between any two points on a line.

Figure %: Finding an infinite number of points on a line

Points are called colinear if they lie in the same line. Likewise, points are called noncolinear if they lie in different lines. Since a line is determined by two points, any two points are always colinear. When a group of three points is considered, however, they may be noncolinear. Colinearity is a relative term. Points are only colinear or noncolinear when considered with respect to other points. The figure below has a set of noncolinear points on the left, and a set of colinear points on the right.

Figure %: Points A, B, and C are noncolinear, whereas points D, E, and F are colinear
Note, as stated above in the rule that any two points are colinear, that a line can be drawn through any of the two points in the diagram. Though point A is noncolinear with respect to D, E, and F, it is colinear with D.

Line Segments

A line segment is the portion of a line that lies between two points on that line, points A and B. Whereas a line has infinite length, a line segment has a finite length. A line segment is denoted by segment AB, or the symbol

Figure %: The symbol for segment AB
Line segments of the same length are called congruent. A dash or a set number of dashes is drawn through congruent segments to symbolize their congruence. Here is a figure of a segment:

Figure %: Segment AB

Rays

A ray is a cross between a line and a line segment. It extends without bound in one direction, but not the other. It is determined by two points, one being the starting point for the ray, and the other determining the direction of the ray. A ray can be symbolized in the following way:

Figure %: The symbol for ray AB
Below is a figure of a ray:
Figure %: Ray AB
Just like lines, segments and rays have no thickness, only length. They are intangible, and only used to specify a set of locations in space.

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