Segment

In geometry, the segment is a fragment of the line that is included between two points, called extreme or final points. Thus, given two points A and B, segment AB is called the intersection of the ray of origin A that contains point B with the ray of origin B that contains point A. Points A and B are endpoints of the segment and the points on the line to which the segment belongs.

Segment AB (green) contained in straight AB.

Consecutive segments

Segments in a row.

Two segments are consecutive when they have only one end in common. Depending on whether or not they belong to the same line, they are classified as:

• Colinears, aligned or adjacent.
• Not colinear.

Segments as quantities

The golden number arises from the division in two of a segment saving the following proportions: The total length a+b is the longest segment aLike, a is the shortest segment b.

The set of metric segments constitutes a magnitude, of which the segments are quantities. It is possible to determine relationships between them and carry out the operations defined for the elements of a quantity:

Segment Comparison

Three Possibilities Postulate (Law of Trichotomy): Given two segments, one and only one of the following three possibilities must hold:

• The segments are the same.
• The first is greater than the second.
• The first is less than the second.

Possibilities that are excluded and completed, that is, when one is fulfilled, the other two are no longer fulfilled.

Segment equality

Segment equality, verifiable by superposition, has the following properties:

• Identic, reflective or reflective: Any segment is equal to itself.
• Reciprocal or symmetrical: If one segment is congruent with another, it is congruent with the first.

Inequality

Segment inequality enjoys the transitive property for relations of greater and lesser.

Operations

The following operations are distinguished:

Sum

The sum of several consecutive collinear segments results in the segment determined by the non-common ends of the considered segments. Geometrically, the sum of segments is another segment that is obtained by constructing segments collinearly orderly congruent with the given ones, and proceeding as indicated at the beginning.

The sum of two segments is another segment that starts at the origin of the first segment and ends at the end of the second segment.

The length of the sum segment is equal to the sum of the lengths of the two segments that form it.

Subtraction

To subtract segments, the subtrahend segment is carried over the minuend segment so that their origins coincide, the segment that remains from the end of the subtrahend to the end of the minuend represents the difference.