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'.

Slope of a Line through Two Given Points

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.

● The Straight Line

• Straight Line
• Slope of a Straight Line
• Slope of a Line through Two Given Points
• Collinearity of Three Points
• Equation of a Line Parallel to x-axis
• Equation of a Line Parallel to y-axis
• Slope-intercept Form
• Point-slope Form
• Straight line in Two-point Form
• Straight Line in Intercept Form
• Straight Line in Normal Form
• General Form into Slope-intercept Form
• General Form into Intercept Form
• General Form into Normal Form
  
• Point of Intersection of Two Lines
  
• Concurrency of Three Lines
• Angle between Two Straight Lines
• Condition of Parallelism of Lines
• Equation of a Line Parallel to a Line
• Condition of Perpendicularity of Two Lines
• Equation of a Line Perpendicular to a Line
• Identical Straight Lines
• Position of a Point Relative to a Line
• Distance of a Point from a Straight Line
• Equations of the Bisectors of the Angles between Two Straight Lines
• Bisector of the Angle which Contains the Origin
• Straight Line Formulae
• Problems on Straight Lines
• Word Problems on Straight Lines
• Problems on Slope and Intercept

11 and 12 Grade Math

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