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Line Segment

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In Geometry, line segment is the part of the line that has a fixed length and two end points. If we talk about a line, it is the group of the points which goes in both the directions endlessly, so it does not have fixed length and the starting and the ending Point. At both the ends of the line, we have an arrow mark, indicating that is goes up to infinite. 

    Line Segment
If we have one point then infinite lines can be drawn from it. On the another hand we say that if we have two points, then only one Line Segment can be drawn through it. Similarly if we have 3 points, then we can frame 3 line segments from 3 points. Thus the figure formed by joining 3 line segments is called a triangle. It is a closed figure with three vertex, three angles and 3 line segments and the angle sum of the triangle so formed is of 180 degrees.

On joining 4 points, we can again draw a closed figure called a quadrilateral. A quadrilateral is formed by joining 2 Triangles, so we also say that the sum of angles of the quadrilateral is 360 degrees. This is equal to the sum of angles of 2 triangles.

A line segment has a fixed length so we are able to find the length of the boundary of the closed figures formed by joining these line segments.  This length sum of the line segments is called the perimeter of the closed figure. If we want to find the perimeter of the closed figures which does not have any curved boundaries, then the perimeter is the sum total of the enclosing line segments of the polygons.

A line segment is the link joining the two points. So we say that the line segment is called either the edge of the polygons or we can even say that the diagonals drawn in the polygons are also line segments. The two points at the end of the two sides of the line segments indicates that the line segment will not extend beyond a particular point.

Open Line Segment

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Before we discuss about open line segment, it is important for us to know something about Line Segment. Line segment and open Line Segment are both Geometry topics. Line segment is a small segment or part of a line which is restricted or bounded by two different endpoints or terminal points. This line segment will have each and every Point of the line between these endpoints.

There are so many examples of line segments: the geometric shapes like Square, trapezium, triangle, parallelogram, rectangle, cube which are finite are bounded between some points, therefore their sides are nothing but line segments. 

The open line segment does not include the endpoints that are placed in the brackets. For a Closed Line Segment parentheses i.e, ( ) are used. An open line segment is defined as the subset of a line. We are denoting this subset as L and this can be parameterized as:
Let x and y is two vectors which belong to: u, v ∈V.
L = u + t v | t  ∈(0,1)

One important point in open line segment is that when in a Polygon, the end points of a line are the vertices of the polygon and if these vertices are adjacent, then that open line segment is called as an edge of that polygon and if they are not adjacent then the open line segment is known as a diagonal.

The equation of a line segment with two endpoints B(bx, by) and D(dx, dy), when there is point C in between B and D such that the distance BC if added to the distance CD is equal to the distance BD, is,
$\sqrt{(x–d_{x})^{2}+(y–d_{y})^{2}}+ \sqrt{ (x–b_{x})^{2}+(y–b_{y})^{2}}=\sqrt{(d_{x}–b_{x})^{2}+(d_{y}–b_{y})^{2}}$

 

Closed Line Segment

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Before jumping on to the topic closed line segment, it is important for us to understand the topic Line Segment which is of course the basis of the topics closed Line Segment. In Geometry, we study about the term line, where infinite points are placed successively in a straight row.

Line segment is bounded between two different extreme points, so we can say that a line segment is just a part of the line and it consists of every Point coming in between those two different points. We have two types of line segments: open line segment and closed line segment.

The closed line segment includes the endpoints that are placed in the brackets. For a closed line segment Square brackets i.e, [ ] are used. An closed line segment is defined as the subset of a line. We are denoting the line segment L as the subset as V and this can be parameterized as:
L = u + t v | t  ∈[0,1]
where x and y are two vectors and u, v ∈V.
 
We have so many examples in our surrounding of different line segments. Generally, in Polygon, if the vertices are the end points of a line segment and they are adjacent to each other, then that line segment is known as an edge. If vertices are not adjacent then it will definitely be a diagonal. If the two end point of a closed line segment lie on the curves like Circle and Semicircle then they are known as the chords of these curves.

Consider a line segment having its end points as A and C and between these two points we have a point on the line segment B, then the sum of the distance AB and BC is equal to the distance AC. We have a equation of line segment with end points A (ax, ay ) and C (cx, cy):
 
$\sqrt{(x–c_{x})^{2}+(y–c_{y})^{2}}+ \sqrt{ (x–a_{x})^{2}+(y–a_{y})^{2}}=\sqrt{(c_{x}–a_{x})^{2}+(c_{y}–a_{y})^{2}}$

Line Segment Symbol

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A line can be defined as straight geometric shape which extends infinitely in any direction in space. Line segment is a part of this geometric shape. It is not like a line which has property of not having termination.

A Line Segment is confined between two end points. These endpoints can be letters which can be used to name or describe the line. For instance, consider the following line in the figure:
Line Segment Symbol
In this figure we have a line whose end points have been marked as “A” and “B”. So line can be named as line AB and length of line can be written as AB = 5 units. AB here is basically a Line Segment symbol, which looks like a Straight Line and is placed above two letters 'A' and 'B' or we can say that a line passes through two points 'A' and 'B' such that a line segment is obtained.

When you are drawing a line segment you need to know the endpoints of it. For instance, consider a line segment having endpoints as 'F' and 'G'. The representation of a line segment symbol can be given by writing both letters together and remember to keep them in uppercase i.e, 'FG'. Enclosure of two endpoints also helps us to represent the length of line segment.

In case we are given coordinates of two points as F(x, y) and G(n, m), we can write the equation as,
Y–y =$\left(\frac{m–y}{n–x}\right)$ * (X–x)