# Line segment Calculator

This tool calculates and plots a line segment graph. Calculated properties include length, midpoint, slope, normal vector and perpendicular bisector. It also calculates its explicit, parametric and vector equations.

## Line segment Formulas

Notations :

S : a line segment in a cartesian plane

(x_{1} , y_{1}) : coordinates of S's starting point

(x_{2} , y_{2}) : coordinates of S's endpoint point

### Line segment length

`L = sqrt((x_2-x_1)^2+(y_2-y_1)^2)`

### Line segment slope

Segment slope is defined when `x_2 != x_1` (non vertical line segment)

`m = (y_2-y_1)/(x_2-x_1)`

### Explicit Equation of line segment

We assume that `x_2 != x_1` (i.e., segment S is not vertical)

m is the slope of segment S defined above,

Let p defines as `p = y_1 - m*x_1`

Then, an explicit equation of segment S can be written as follows:

`y = mx+p , x in [x_1,x_2]`

We assume that `x_1 < x_2` otherwise, replace `[x_1,x_2]` by `[x_2,x_1]`.

### Vector equation of a line segment

Segment S consists of all points M such that:

`\vec(OM) = \vecu + t \vecv \text{ } t in [0,1]` ,

`\vecu = ( x_1 , y_1 )`

`\vecv = ( x_2-x_1 , y_2-y_1 )`

### Parametric equation of a line segment

This equation is derived from the vector equation (see above). A point M with coordinates (x, y) belongs to the line segment S if and only if,

`x = x_1 + t*(x_2-x_1)`

`y = y_1 + t*(y_2-y_1)`

`t in [0,1]`

### Midpoint of a line segment

The coordinates of the midpoint of segment S are as follows,

`x_m = (x_1+x_2)/2`

`y_m = (y_1+y_2)/2`

### Perpendicular bisector of a line segment

We assume the following:

* `y_2 != y_1` and `x_2 != x_1` (i.e. segment S is neither horizontal nor vertical)

* m : slope of segment S (defined above)

* (`x_m` , `y_m`) : coordinates of the midpoint of S (defined above).

Then, the equation of the perpendicular bisector is:

y = n.x + q

Where n and q represent the slope and y-intercept, respectively. They are defined as:

`n = -1/m`

`q = y_m - n*x_m`

### Normal vector of a segment

The Normal vector `\vecn` of segment S is orthogonal to the vector that originates from the starting point and ends at the endpoint of S,

`\vecn` : (`y_1-y_2` , `x_2-x_1`)

## See also

Line Calculator

Vector Calculator

Coordinate Geometry calculators

Geometry calculators

Mathematics calculators