A line segment can be defined as a part of line that is bounded by two points. The points are called the end points of the line segment.

Consider a line, denoted by L. Let A and B be two points on the line. Then, the line segment betweenA and B, denoted by $\overline{AB}$, contains the end points A and B, every point on the line between A and B.

## Types of Line Segments

Line Segments can represent the sides of a triangle or a polygon and the diagonals. On the basis of it, the line segments can be of the following types:
• Diagonal Line Segment
• Horizontal Line Segment
• Vertical Line Segment

Here is the description of all the above mentioned types of line segments:

Diagonal Line Segment: The line segment that cuts diagonally from one corner of a square or rectangle to its opposite corner is called the Diagonal Line Segment. The slope of a diagonal line segment can be positive or negative and it cuts the square or rectangle into two congruent triangles.

In the above figure, $\overline{AD}$ is a diagonal line segment with positive slope and it partitions the rectangle $\square$ABDC into two congruent triangles $\Delta$ABD and $\Delta$DCA; and $\overline{BC}$ is a diagonal line segment with negative slope and it partitions the rectangle $\square$ABDC into two congruent triangles $\Delta$ABC and $\Delta$DCB.

Horizontal Line Segment: Horizontal Line Segment is a line segment in the Coordinate Plane with all points on the line segment share the same y-coordinate. In the figure, $\overline{AB}$ is a horizontal line segment.

Vertical Line Segment: Vertical Line Segment is a line segment in the Coordinate Plane, with all points on the line segment share the same x-coordinate. In the figure, $\overline{AC}$ is a vertical line segment.

## Length of a Line Segment

Since a line can be extended indefinitely on both sides, a line can have slope but its length can’t be determined. We can say that a line does have infinite length. But, a line segment has end points and hence a line segment has length. The length of a line segment can be drawn using a straight edge with marked units of measurement.

The length of the line segment $\overline{AB}$ is denoted by AB.
In the above figure, we have AB=6 cm. However, measuring the length of the line segment with a straight edge is not accurate and is approximately equal. However, in the coordinates of the end points known, it is possible to measure the length of the line segment accurately by using the distance formula, as explained hereunder.

Consider a line segment $\overline{AB}$ with the end-points A(x1, y1) and B(x2, y2). Let the projection of $\overline{AB}$ on the x-axis be $\overline{CD}$. Then, it is clear that C(x1, 0) and D(x2, 0). Join $\overline{AC}$ and $\overline{BD}$. Draw a perpendicular from A onto BD and suppose that it intersects $\overline{BD}$ at E. Then, the coordinates of E are given by (x2, y1). From the figure, it is clear that
AE = CD = OD - OC = x2 - x1
BE = BD - DE = y2 - y1
Since $\Delta$AEB is a right triangle, by Pythagorean Theorem,
Hypotenuse2 = Side2 + Side2
$\rightarrow$AB2 = AE2 + BE2
$\rightarrow$AB2 = (x2 - x1 )2 + (y2 - y1 )2
$\rightarrow$AB2 = (x1 - x2 )2 + (y1 - y2 )2
$\rightarrow$AB = $\sqrt{(x_1 - x_2 )^2 + (y_1 - y_2 )^2}$

### Solved Example

Question: Given a line segment $\overline{AB}$ with end points A(-2, 3) and B(3, 4). Find the length AB of the line segment $\overline{AB}$.
Solution:

Given that,
A(x1, y1) = (-2, 3) and B(x2, y2) = (3, 4)
Thus, we have
x1 = -2, y1 = 3, x2 = 3, and y2 = 4
Now,
Length of $\overline{AB}$  = AB
= $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$
= $\sqrt{(-2 - 3)^2 + (3 - 4)^2}$
= $\sqrt{(-5)^2 + (-1)^2)}$
= $\sqrt{25+1}$
= $\sqrt{26}$ units.

## Mid-point of a Line Segment

Consider a line segment $\overline{AB}$ with end points A($x_1$, $y_1$ ) and B($x_2$, $y_2$). Let M(x, y) be a point on the line segment $\overline{AB}$ which divides AB in m:n ratio. Let C, D, and E be the feet of perpendiculars of A, B, and M, respectively. Join AC, ME, and BD. Let F be the foot of the perpendicular of A on ME, and G be the foot of the perpendicular of M on BD.

From the figure, it is clear that
AF = $x - x_1$, MF = $y - y_1$, MG = $x_2 - x$, and BG = $y_2 - y$
Moreover, $\Delta$AFM and $\Delta$MGB are similar triangles. Thus, we have $\frac{AM}{MB}$ = $\frac{AF}{MG}$ = $\frac{MF}{BG}$$\rightarrow$$\frac{m}{n}$ = $\frac{x-x_1}{x_2 - x}$ = $\frac{y - y_1}{y_2 - y}$

$\rightarrow$m($x_2 - x$) = n($x - x_1$ ) and m($y_2 - y$) = n($y - y_1$)

$\rightarrow$m$x_2 - mx$ = $nx-nx_1$ and m$y_2$ - my = $ny - ny_1$

$\rightarrow$(m+n)x = $mx_2$ + $nx_1$ and (m+n)y = $my_2$ + $ny_1$

$\rightarrow$x = $\frac{mx_2 + nx_1}{m+n}$ and y = $\frac{my_2 + ny_1}{m+n}$

So, we have

M(x, y) = ($\frac{mx_2 + nx_1}{m+n}$, $\frac{my_2 + ny_1}{m+n}$)

If M is the mid-point of $\overline{AB}$, then m=n. Thus,
Mid-point of $\overline{AB}$ = ($\frac{x_1 + x_2}{2}$, $\frac{y_1 + y_2}{2}$)

### Solved Example

Question: Find the mid-point of the line segment $\overline{AB}$ with end point A(-3, 4) and B(7, 2).

Solution:

Given that
A($x_1$, $y_1$) = (-3, 4)
B($x_2$, $y_2$) = (7, 2)
Thus, the mid-point of $\overline{AB}$ is given by

M = ($\frac{x_1 + x_2}{2}$, $\frac{y_1 + y_2}{2}$)

=($\frac{-3+7}{2}$, $\frac{4+2}{2}$)

=($\frac{4}{2}$, $\frac{6}{2}$)

=(2, 3)

## Parallel Line Segments

Two line segments $\overline{AB}$ and $\overline{CD}$ are called parallel line segments if the lines containing them never intersect with each other. If two line segments are parallel, they incline the same angle with a horizontal line.

If two line segments are parallel, the lines containing them make the angle 0° with each other.

## Perpendicular Line Segments

Two line segments are said to be perpendicular, if the lines containing them incline at an angle of 90°. It is to be that two perpendicular line segments need not intersect each other, but the lines containing them intersect at right angles.

In the figure $\overline{AB}$ and $\overline{CD}$ are perpendicular line segments, but they are not intersecting each other. Indeed, the lines containing them are intersecting each other at right angles.

## Naming Line Segments

A line segment with end point A and B, is denoted by $\overline{AB}$. The order of the end points in the notation is not important and hence we can also denote it as $\overline{BA}$. For example, if we consider a line with three points A, B, and C as shown in the figure. Then, we can have the following possible line segments.

• The Line Segment $\overline{AB}$ with end the points A and B.
• The Line Segment $\overline{AC}$ with end the points A and C.
• The Line Segment $\overline{BC}$ with the end points B and C.

We denote the length of a line segment $\overline{AB}$ by AB or |$\overline{AB}$|.

## Line Segment Bisector

Line Segment Bisector is the locus of all those points which are equidistant from both the end-points. It is to be noted that the locus of all such points is a straight line that intersects the given line segment at right angle and passes through the mid-point of the line segment, as shown in the following figure.

Observe that P is a point on the Line Segment Bisector. Thus, it is equidistant from the end points A and B of the line segment $\overline{AB}$. It can also be observed that the locus of all such points is a straight line.

## Skew Line Segments

On a plane, the lines containing two parallel line segments never intersect each other. If two line segments are non-parallel, the lines containing them always intersect with one other. But, in a three dimensional space even if the lines are non-parallel they may not intersect each other.

Line segments on such lines are called skew line segments. Since the skew lines are not parallel, they are not coplanar as the lines on the same plane either intersect or parallel.
It is to be noted that skew lines exist only in three or multi-dimensional spaces.

## Intersecting Line Segments

Lines containing two parallel line segments never intersect each other, but the lines containing two non-parallel line segments will always intersect each. But, the intersection point may or may not be on the given line segments.

## Examples of Line Segments

### Ray Line

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