Line segment is a term that comes under geometry and has two end points that are distinct in nature. Between the two end points, all the other points fall in a straight line. Now under the term line segment, there are many sub-terms that are commonly used such as closed line segment that has both the endpoints. On the other hand, you will come across a term called open line segment that does not include the two endpoints. There is a third term called the half-line segment where we consider only one out of the two endpoints.
Some of the common examples of line segments that you will come across are the sides of a triangle or that of a square. There are many more geometrical figures such as the rhombus and rectangles which have line segments in them. On the whole, pretty much all of the figures in geometry except the circle and a few more shapes which have curves in them do not have line segments. The rest are compositions of line segments arranged in an orderly manner to form a distinct shape. It is a well-known fact in the world of coordinate geometry that if the endpoints of a line segment happen to be the vertices of any polyhedron or polygon, it will be called an edge provided that the vertices in question are adjacent to each other. Otherwise, it will be called a diagonal. Suppose that the endpoints of a segment fall on a curve such as in a circle, then it will be called as a chord belonging to that particular curve.
Now we come to the part where we define the difference between ray, a line and line segment. Going according to the definition, a line is defined as a collection of continuous and adjacent points which can have two or more points. On the other hand, a ray is defined as a straight line at a particular point which tends to be aligned towards one direction. The design of these three can also be used to understand the difference. If you notice all three side by side, you will realise that for a ray, one end will be an arrow pointing towards a direction whereas the other end will be an endpoint. A line segment has two endpoints whereas a line will have arrows on both ends pointing in opposite directions. One more point to note about line segment is that the limit of the length does not go till infinity. Hence, it is always finite.
There are lots of ways in which line segments can be defined. It is basically a non-empty set that is connected. Let us consider V as topological vector space. In such a case, there will be a closed line segment which will be a closed set within V. On the other hand, an open set will be an open line segment, but only when V is one-dimensional in nature. The segment of line concepts generally come under the ordered geometry domain. Take note of the fact that line segments can take multiple forms such as parallel, intersecting, skew or it might not be any of these. Let us say if there are non-parallel lines that exist in one Euclidean plane, then they will have to cross each other, but this is something which is not valid for other segments.
In a coordinate system, there is a concept known as axiomatic treatment and the notion for the same is basically an assumption which is used to satisfy many axioms related to the isometry of a line as well as a line segment. Segments have a vital role to play in many other theories. Let us consider some of the examples related to the properties of line segments.
Considering that a particular geometrical set is convex in nature and if there is a segment that brings two points together, then it justifies the fact that the figure is convex. This might be a complicated concept, but it throws light on the analysis of convex sets and correlates it to line segment analysis. There is a segment addition postulate which is widely followed in coordinate geometry and can also be used for adding congruent segments that have equal lengths. In this way, other segments are consequently substituted and this makes the collection of segments congruent.
Let us discuss the concept of directed line segment. When there is a particular orientation assigned to a line segment, it will point towards a particular direction and will be considered as ray. This is a form of translation and we are giving a direction to the segment altogether. Hence, all of the points that are falling in the segment will be alignment along the orientation that we have fixed. The direction and the magnitude of a vector are always indicative towards a potential change. This is a suggestion that was gradually absorbed deep into mathematical physics since it was in a direction correlation of the Euclidean vector. So if any pair has the same length as well as the same orientation, then we usually reduce it and this is done by making those pairs equivalent of all of the directed line segments that are part of the collection. All this is directly related to the concept of equipollence of line segments introduced by Guisto Bellavitis in 1835.
Line segments appear as edges as well as diagonals of polygons, but there are other forms of shapes as well which involve line segments. Let us go through some of these shapes in no order.
Triangles have three altitudes where each of the sides perpendicularly connects another side and it could be an extension as well, to the vertex directly opposite to it. Apart from that, there are three medians and each one of them connects the midpoint of a side perpendicularly with the vertex right opposite to it. There are perpendicular bisectors on each side which connect midpoints to opposite sides and there are internal angle bisectors as well which also connect the mid-points to the vertex of opposite sides. There are multiple equalities related to the lengths of these segments and there are various inequalities as well. Many other segments of a triangle include the centres. There are many centres in a triangle namely circumcenter, incenter, nine-point centre, orthocenter and centroid.
Next up, we will discuss quadrilaterals where there is a high involvement of the concept of a line segment. There are sections of a quadrilateral such as bi-medians which also connect the middle points of each of the sides with the middle points of the opposite sides. In addition to that, there are four maltitudes which perpendicularly connect the midpoints with sides.
Finally, we come to ellipses and circles. Although towards the beginning of the article, we emphasised on the fact that there is barely any involvement of line segment in a circle since the whole circumference is a curve. And the basic concept of a line is that each of the points has to make the same angle. But yes, that does not mean that there is absolutely no involvement of line segment concept in a circle. If there is a line segment that connects any two points inside an ellipse or a circle, then it is known as a chord. A diameter of a circle is ideally the longest chord that is present since it intersects the centre of the circle. If there is a line segment that connects any point on the circumference to the centre of the circle, then it is known as the radius of the circle. The concept is somewhat the same for that of an ellipse since the longest chord is known as the major axis. Now if there is a segment that emerges from the midpoint of the major axis, i.e. centres of the ellipse and extends to either one of the endpoints is known as semi-major axis. In the same way, the minor axis is the term that we use to denote the shortest chord of the ellipse and the same definition of semi-major axis applies for the semi-minor axis as well, although the context would be opposite of the former. Also, there is a concept of latera recta of the ellipse which is when the chords of the ellipse which are perpendicular towards the major axis of the ellipse and tend to cross either one of its foci.