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.

10th Grade Math

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