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.

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 The Straight Line





11 and 12 Grade Math

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