Slope of the Line Joining Two Points

We will discuss here about the slope of the line joining two points.

To find the slope of a non-vertical straight line passing through two given fixed points:

Let P (x$_{1}$, y$_{1}$) and Q (x$_{2}$, y$_{2}$) be the two given points. According to the problem, the straight line PQ is non-vertical x$_{2}$ ≠ x$_{1}$.

Required to find, the slope of the line through P and Q.

From P, Q draw perpendiculars PM, QN on x-axis and PL ⊥ NQ. Let θ be the inclination of the line PQ, then ∠LPQ = θ.

Slope Formula

From the above diagram, we have

PL = MN = ON - OM = x$_{2}$ - x$_{1}$ and

LQ = = NQ - NL = NQ - MP = y$_{2}$ - y$_{1}$

Therefore, the slope of the line PQ = tan θ

                                                  = $\frac{LQ}{PL}$

                                                  = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

                           = $\frac{Difference\, of\, ordinates\,of\, the\, given\, points}{Difference\, of\, their\, abscissae}$

Hence, the slope (m) of a non-vertical line passing through the points P (x$_{1}$, y$_{1}$) and Q (x$_{2}$, y$_{2}$) is given by

slope = m = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

1. Find the slope of the line passing through the points M (-2, 3) and N (2, 7).

Solution:

Let M (-2, 3) = (x$_{1}$, y$_{1}$) and N (2, 7) = (x$_{2}$, y$_{2}$)

We know that the slope of a straight line passing through two points (x$_{1}$, y$_{1}$) and (x$_{2}$, y$_{2}$) is

m = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Therefore, slope of MN = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$ = $\frac{7 - 3}{2 + 2}$ = $\frac{4}{4}$ = 1.

2. Find the slope of the line passing through the pairs of points (-4, 0) and origin.

Solution:

We know that the coordinate of the origin is (0, 0)

Let P (-4, 0) = (x$_{1}$, y$_{1}$) and O (0, 0) = (x$_{2}$, y$_{2}$)

We know that the slope of a straight line passing through two points (x$_{1}$, y$_{1}$) and (x$_{2}$, y$_{2}$) is

m = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

Therefore, slope of PO = $\frac{y_{2} - y_{1}}{x_{2} - x_{1}}$

                                 = $\frac{0 - (0}{0 - (- 4)}$

                                 = $\frac{0}{4}$

                                 = 0.

● Equation of a Straight Line

• Inclination of a Line
• Slope of a Line
• Intercepts Made by a Straight Line on Axes
• Slope of the Line Joining Two Points
• Equation of a Straight Line
• Point-slope Form of a Line
• Two-point Form of a Line
• Equally Inclined Lines
• Slope and Y-intercept of a Line
• Condition of Perpendicularity of Two Straight Lines
• Condition of parallelism
• Problems on Condition of Perpendicularity
• Worksheet on Slope and Intercepts
• Worksheet on Slope Intercept Form
• Worksheet on Two-point Form
• Worksheet on Point-slope Form
• Worksheet on Collinearity of 3 Points
• Worksheet on Equation of a Straight Line

10th Grade Math

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