Finding the Slope of an Equation of a Line

We have seen in geometry there are different types of lines drawn on the coordinate planes. But how can we know whether the drawn lines are perpendicular to each other or parallel to each other or make an angle? This is possible only if we can measure the slope of the lines. Thus, you may have understood the importance of slope in geometry; let’s discuss the definition of slope, the formula of slope for a different line like parallel, perpendicular and collinear. Also, we will discuss measuring the angles between lines in terms of the slope. 

Definition of the slope

The slope of the line tells us about the inclination and the direction of the line. It is measured by taking the ratio of the change in the x-coordinates and y-coordinates of the line. We can write it in an equation in the below form,

Slope =m = (y2 – y1 )/ (x2 -x1)

Where m is the slope and (y2 – y1 )and (x2 -x1) change the y and x- coordinates, respectively.

The slope of a line

A slope of a line is given by

y = mx + c

Where m is a slope of the line, x is x-coordinate, y is y-coordinate, and c is the y-intercept.

The slope is also denoted by tan θ, where θ is the angle with the positive x-axis made by the line.

Finding the Slope of an Equation of a Line

The slope of the parallel lines

Parallel lines have an equal slope. They have the same inclination with the positive x-axis, and thus their slope is measured concerning the x-axis. Slopes of parallel lines are equal. If we have a line with slope m1 and the other line with slope m2, and as per the condition given, they are parallel to each other, then m1=m2. The equation represents parallel, like having an equal coefficient of x and y.

The slope of the perpendicular line

The intersection of lines by making an angle of 90 degrees are called perpendicular lines. When drawn on the coordinate plane, the perpendicular lines make a slope with the angle in such a way that the slope of one line is a negative reciprocal of the slope of the other line. For example, let’s take the two perpendicular lines with the m1 and m2. The relationship between the two perpendicular lines can be represented by m1.m2= -1, i.e. the product of the slope of the two perpendicular lines will be -1.  

Thus, we can write it as m1= -1/m2

The slope of the collinear points

The point through which a single straight line can pass is collinear points. The slope formula of collinear points is useful in finding out the nature of points on the coordinate plane, like if they are in a straight line or not, i.e. to check their collinearity. If all three points have an equal slope, then the lines formed by them are said to be collinear. For example, if we have three points, a, b, and c, collinear, then the slope of the ab=slope of line bc= slope of line ac. The following formula is used to calculate the slope of the line,

Slope =m = (y2 – y1 )/ (x2 -x1) where(x1,y1) and (x2,y2) can be coordinates of any two points

The angle between two lines

If the two lines except for the perpendicular and the parallel lines (since we know their angles) intersect at a point, they lead to the formation of an angle. This angle can be expressed in the term of the slope of the line using the following formula,

The slope of the vertical and horizontal lines

Lines that are vertical to each other do not have any steepness towards the x-axis; therefore, the vertical line will have no value for the axis. We can also say that we cannot get the inclination of the line.

By using the formula of the slope,

Slope =m =  (y2 – y1 )/ (x2 -x1) , and putting x1 and x2 = for the vertical lines we get,

m= (y2 – y1 )/0 = undefined.

In the case of the horizontal line, the y-axis coordinate will be 0.

Therefore, by slope formula, the horizontal line’s slope becomes 0.

Conclusion

Thus, we can conclude the following points from the above descriptions.The formula of the  slope of the line= Slope =m =  (y2 – y1 )/ (x2 -x1) .The slope of the parallel lines are equal, that is m1=m2

The product of the slope of the two perpendicular lines will be -1 since one slope is the reciprocal of the other.

If three points a, b, and c are collinear, then the slope of the ab =slope of line bc = slope of line ac. Horizontal lines have zero slopes, and vertical lines have an undefined slope.