If the Slopes of Two Lines Are Equal
In geometry, we have seen the lines fatigued on the coordinate plane. To predict whether the lines are parallel or perpendicular or at any bending without using whatsoever geometrical tool, the all-time way to find this is past measuring the slope. In this article, nosotros are going to discuss what a gradient is, slope formula for parallel lines, perpendicular lines, slope for collinearity with many solved examples in detail.
What is a Slope?
In Mathematics, a slope of a line is the alter in y coordinate with respect to the modify in ten coordinate.
The net change in y-coordinate is represented by Δy and the cyberspace change in ten-coordinate is represented by Δx.
Hence, the change in y-coordinate with respect to the alter in x-coordinate is given past,
m = alter in y/alter in x = Δy/Δx
Where "m" is the slope of a line.
The slope of the line can also be represented past
tan θ = Δy/Δx
And so, tan θ to be the slope of a line.
Mostly, the slope of a line gives the mensurate of its steepness and management. The slope of a directly line between two points says (x 1 ,y 1 ) and (10 2 ,y 2 ) tin be hands determined by finding the difference between the coordinates of the points. The slope is ordinarily represented by the letter 'm'.
Slope Formula
If P(x1,y1) and Q(102,yii) are the two points on a straight line, then the slope formula is given by:
m = (y 2 – y ane )/(x 2 – ten 1 )
Therefore, based on the in a higher place formula, we can easily summate the gradient of a line betwixt two points.
In other term, the slope of a line between 2 points is as well said to exist the rising of the line from one point to another (forth y-centrality) over the run (along x-centrality). Therefore,
Slope, m = Rise/Run
Slope of a Line Equation
The equation for the slope of a line and the points besides chosen point slope form of equation of a direct line is given by:
Whereas the slope-intercept course the equation of the line is given by:
y = mx + b
Where b is the y-intercept.
How to Find Slope of a Line on a Graph?
In the given figure, if the angle of inclination of the given line with the ten-axis is θ then, the slope of the line is given by tan θ. Hence, at that place is a relation between the lines and angles. In this commodity, you lot volition larn diverse formulas related to the angles and lines.
The slope of a line is given as m = tan θ. If two points A \((x_1,y_1)\) and B \((x_2,y_2) \) lie on the line with (\( x_1\) ≠ \(x_2\)) and then the slope of the line AB is given every bit:
1000 = tan θ = \( \frac{y_2~-~y_1}{x_2~-~x_1}\)
Where θ is the angle which the line AB makes with the positive direction of the 10-centrality. θ lies betwixt 0° and 180°.
It must be noted that θ = 90° is only possible when the line is parallel to y-axis i.eastward. at \( x_1 \) = \( x_2,\) at this particular angle the slope of the line is undefined.
Conditions for perpendicularity, parallelism, and collinearity of straight lines are given below:
Slope for Parallel Lines
Consider 2 parallel lines given by \( l_1\) and \( l_2 \) with inclinations α and β respectively. For 2 lines to be parallel their inclination must also exist equal i.due east. α=β. This results in the fact that tan α = tan β. Hence, the condition for two lines with inclinations α, β to exist parallel is tan α = tan β.
Therefore, if the slopes of two lines on the Cartesian airplane are equal, then the lines are parallel to each other.
Thus, if two lines are parallel and then, \( m_1\) = \( m_2 \) .
Generalizing this for n lines, they are parallel merely when the slopes of all the lines are equal.
If the equation of the 2 lines are given equally ax + by + c = 0 and a' ten + b' y + c'= 0, then they are parallel when ab' = a'b. (How? You can get in at this effect if y'all find the slopes of each line and equate them.)
Slope for Perpendicular Lines
In the figure, we have two lines \( l_1\) and \( l_2 \) with inclinations α, β. If they are perpendicular, we tin can say that β = α + 90°. (Using properties of angles)
Their slopes can be given as:
10001 = tan(α + 90°) and \( m_2 \) = \( tan~α\).
⇒\( m_1 \) = – cot α = \( -~ \frac{1}{tan~\alpha}\) = \( -~\frac {1}{m_2}\)
⇒\( m_1 \) = \( -\frac {1}{m_2} \)
⇒\( m_1 ~×~m_2 \) = -1
Thus, for two lines to be perpendicular the production of their slope must exist equal to -1.
If the equations of the ii lines are given by ax + by + c = 0 and a' 10 + b' y + c' = 0, and so they are perpendicular if, aa'+ bb' = 0. (Again, you lot can arrive at this upshot if y'all find the slopes of each line and equate their product to -i.)
Too, read: Perpendicular Lines
Slope for Collinearity
For two lines AB and BC to be collinear the slope of both the lines must exist equal and there should be at least one common point through which they should pass. Thus, for three points A, B, and C to be collinear the slopes of AB and BC must be equal.
If the equation of the 2 lines is given past ax + past + c = 0 and a' ten+b' y+c' = 0, then they are collinear when ab' c' = a' b' c = a'c'b.
Angle between Two Lines
When two lines intersect at a betoken and so the angle between them tin be expressed in terms of their slopes and is given past the post-obit formula:
tan θ = |\( \frac{ m_2~-~m_1}{one~+~m_1~ m_2}\)| , where \( m_1 ~and~ m_2\) are the slopes of the line AB and CD respectively.
If \( \frac{ m_2~-~m_1}{1~+~m_1~ m_2}\)is positive then the bending betwixt the lines is astute. If \( \frac{ m_2~-~m_1}{1~+~m_1~ m_2}\) is negative then the angle betwixt the lines is birdbrained.
Slope of Vertical Lines
Vertical lines have no slope, as they practise not have any steepness. Or it tin be said, we cannot ascertain the steepness of vertical lines.
A vertical line will accept no values for x-coordinates. So, as per the formula of slope of the line,
Slope, m = (y 2 – y one )/(ten ii – x 1 )
Only for vertical lines, x 2 = x 1 = 0
Therefore,
m = (y two – y 1 )/0 = undefined
In the same way, the slope of horizontal line is equal to 0, since the y-coordinates are null.
yard = 0/(x two – ten 1 ) = 0 [for horizontal line]
Positive and Negative Slope
If the value of slope of a line is positive, it shows that line goes upwards as we motility forth or the rise over run is positive.
If the value of slope is negative, and so the line goes done in the graph as nosotros movement along the x-axis.
Solved Examples on Slope of a Line
Example one:
Discover the slope of a line betwixt the points P = (0, –i) and Q = (four,1).
Solution:
Given,the points P = (0, –1) and Q = (4,ane).
As per the gradient formula we know that,
Slope of a line, m = (y 2 – y one )/(10 two – ten ane )
m = (1-(-1))/(four-0) = 2/4 = ½
Example 2:
Observe the slope of a line betwixt P(–2, three) and Q(0, –1).
Solution:
Given, P(–2, 3) and Q(0, –1) are the two points.
Therefore, slope of the line,
m = (-1-three)/0-(-ii) = -4/2 = -2
Example three:
Ramya was checking the graph, and she realized that the raise was 10 units and the run was 5 units. What should be the the slope of a line?
Solution:
Given that, Enhance = ten units
Run = 5 units.
Nosotros know that the slope of a line is defined every bit the ratio of heighten to the run.
i.e. Slope, thou = Heighten/Run
Hence, slope = 10/five = 2 units.
Therefore, the gradient of a line is two units.
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Frequently Asked Questions on Slope of a Line
What is the slope of a direct line?
The gradient expresses the steepness and management of the line. It represents how steep a line is.
What are the three different ways to find the slope?
The slope can be found using unlike methods, such as standard form, slope-intercept form, and point-intercept form.
What is the point-slope equation of a directly line?
The point-slope course of the equation of a direct line is given by:
y − y1 = thou(x − x1)
How to find gradient of a line?
Nosotros demand to find the ratio of the difference between the y-coordinates and x-coordinates of the two points, that form the line. The resulted value is the slope of the line. Information technology shows the rise of the line along the y-axis over the run along x-axis.
What is the slope betwixt ii points?
The gradient between ii points is calculated by evaluating the modify in y-coordinate values and x-coordinate values. For example, the slope between the points (four,8) and (-7,1) is equal to:
m = (1-8)/(-7-4) = -vii/-11 = 7/11
What is the gradient of the line: y = −2x + 7?
The gradient of the line whose equation is y = -2x + seven is -2.
m = -2
Source: https://byjus.com/maths/slope-of-line/
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