- Slope of a Line Or Gradient of a Line.
- Parallelism of Line
- Perpendicularity of Line in Term of Slope
- Collinearity of Points
- Slope of a line when coordinates of any two points on the line are given
- Conditions for parallelism and perpendicularity of lines in terms of their slopes
- Angle between two lines
- Collinearity of three points
If θ is the inclination of a line l, then tan θ is called the slope or gradient of the line l. The slope of a line whose inclination is 90° is not defined. The slope of a line is denoted by m. Thus, m = tan θ, θ ≠ 90°
It may be observed that the slope of x-axis is zero and slope of y-axis is not defined.
Slope of a line when coordinates of any two points on the line are given:
To find the slope of a line in terms of the coordinates of two points on the line.
Let P`(x_1, y_1)` and Q`(x_2, y_2)` be two points on non-vertical line l whose inclination is θ. Obviously, x_1 ≠ x_2, otherwise the line will become perpendicular to x-axis and its slope will not be defined. The inclination of the line l may be acute or obtuse. Let us take these two cases.
Draw perpendicular QR to x-axis and PM perpendicular to RQ as shown in
Case 1 : When angle θ is acute:
∠MPQ = θ . ... (1)
Therefore, slope of line l = m = tan θ.
But in triangle MPQ, we have tan `theta = (MQ)/(MP) `= `(y_2 -y_1)/(x_2 - x_1)` ....(2)
From equations (1) and (2), we have
m = `(y_2-y_1)/(x_2 -x_1)`
Case 2: When angle θ is obtuse:
we have ∠MPQ = 180° – θ.
Therefore, θ = 180° – ∠MPQ.
Now, slope of the line l
m = tan θ
= tan ( 180° – ∠MPQ) = – tan ∠MPQ
=`(-(MQ)/(MP))` =` (-y_2-y_1)/(x_1-x_2) =(y_2-y_1)/(x_2-x_1)`
Consequently, we see that in both the cases the slope m of the line through the points
`(x_1, y_1)` and `(x_2, y_2)` is given by m =`(y_2-y_1)/(x_2-x_1)`.
Conditions for parallelism and perpendicularity of lines in terms of their slopes:
In a coordinate plane, suppose that non-vertical lines `l_1` and `l_2` have slopes `m_1` and `m_2`, respectively. Let their inclinations be α and β, respectively. If the line `l_1` is parallel to `l_2`.
then their inclinations are equal, i.e.,
α = β, and hence, tan α = tan β
Therefore `m_1` = `m_2`, i.e., their slopes are equal.
Conversely, if the slope of two lines `l_1` and` l_2` is same, i.e.,
`m_1` =` m_2` Then tan α = tan β.
By the property of tangent function (between 0° and 180°), α = β. Therefore, the lines are parallel.
Hence, two non vertical lines `l_1` and `l_2` are parallel if and only if their slopes are equal.
Angle between two lines :
Let `L_1` and` L_2` be two non-vertical lines with slopes `m_1` and `m_2`, respectively. If `α_1` and `α_2` are the inclinations of lines `L_1` and `L_2`, respectively. Then
`m_1` = tan`α_1` and `m_2` = tan `α_2`.
two lines intersect each other, they make two pairs of vertically opposite angles such that sum of any two adjacent angles is 180°. Let θ and φ be the adjacent angles between the lines `L_1` and `L_2`. Then θ = `α_2` –` α_1` and `α_1`, `α_2` ≠ 90°.
Therefore tan θ = tan `(α_2` – `α_1`) = `(tanα_2 - tanα_1)/( 1+tanα_1tanα _2)` = `(m_2-m_1)/(1+m_1m_2)` (as 1 + m1m2 ≠ 0)
and φ = 180° – θ so that
tan φ = tan (180° – θ ) = – tan θ = `-(m_2-m_1)/(1+m_1m_2)` , as 1 + m1m2 ≠ 0
Collinearity of three points:
We know that slopes of two parallel lines are equal. If two lines having the same slope pass through a common point, then two lines will coincide. Hence, if A, B and C are three points in the XY-plane, then they will lie on a line, i.e., three points are collinear. if and only if slope of AB = slope of BC.