Slope of a Line through Two Given Points

How to find the slope of a line through two given points?

Let (x\(_{1}\), y\(_{1}\)) and (x\(_{2}\), y\(_{2}\)) be two given cartesian co-ordinates of the point A and B respectively referred to rectangular co-ordinate axes XOX' and YOY'.

Again let the straight line AB makes an angle θ with the positive x-axis in the anticlockwise direction. Now by definition, the slope of the line AB is tan θ.

Therefore, we have to find the value of m = tan θ.

Draw AE and BD perpendiculars on x-axis and from B draw BC perpendiculars on AE. Then,

AE = y\(_{1}\), BD = y\(_{2}\), OE = x\(_{1}\) and OD = x\(_{2}\)

Therefore, BC = DE = OE - OD = x\(_{1}\) - x\(_{2}\)  

Again, AC = AE - CE = AE - BD = y\(_{1}\) - y\(_{2}\)

<ABC = θ, since, BC parallel to x-axis.

Therefore, from the right angle ∆ABC we get,

tan θ = \(\frac{AC}{BC}\) = \(\frac{y_{1} - y_{2}}{x_{1} - x_{2}}\)

⇒ tan θ = \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\)

Therefore, the required slop of the line passing through the points A (x\(_{1}\), y\(_{1}\)) and B (x\(_{2}\), y\(_{2}\)) is

m = tan θ = \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\) = \(\frac{\textrm{Difference of ordinates of the given point}}{\textrm {Difference of abscissa of the given point}}\)


Solved example to find the slope of a line passes through two given points:

Find the slope of a straight line which passes through points (-5, 7) and (-4, 8).

Solution:

We know that the slope of a straight line passes through two points (x\(_{1}\), y\(_{1}\)) and (x\(_{2}\), y\(_{2}\)) is given by m = \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\). Here the straight line passes through (-5, 7) and (-4, 8). Therefore, the slope of the straight line is given by m = \(\frac{8 - 7}{-4 - (-5) }\) = \(\frac{1}{-4 + 5}\) = \(\frac{1}{1}\) = 1

 

Note:

1. Slop of two parallel lines are equal.

2. Slope of x-axis or slope of a straight line parallel to x-axis is zero, since we know that tan 0° = 0.

3. Slop of y-axis or slope of a straight line parallel to y-axis is undefined, since we know that tan 90° is undefined.

4. We know that co-ordinate of the origin is (0, 0). If O be the origin and M (x, y) be a given point, then the slope of the line OM is \(\frac{y}{x}\).

5. The slop of the line is the change in the value of ordinate of any point on the line for unit change in the value of abscissa.





11 and 12 Grade Math

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