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Calculation of the gradient line

Module by: Umeshree Govender, Free High School Science Texts Project. E-mail the authors

Analytical Geometry; Calculation of the gradient line

The gradient of a line describes how steep the line is. In the figure, line PTPT is the steepest. Line PSPS is less steep than PTPT but is steeper than PRPR, and line PRPR is steeper than PQPQ.

Figure 1
Figure 1 (MG10C14_017.png)

The gradient of a line is defined as the ratio of the vertical distance to the horizontal distance. This can be understood by looking at the line as the hypotenuse of a right-angled triangle. Then the gradient is the ratio of the length of the vertical side of the triangle to the horizontal side of the triangle. Consider a line between a point AA with co-ordinates (x1;y1)(x1;y1) and a point BB with co-ordinates (x2;y2)(x2;y2).

Figure 2
Figure 2 (MG10C14_018.png)

So we obtain the following for the gradient of a line:

Gradient=y2-y1x2-x1Gradient=y2-y1x2-x1

We can use the gradient of a line to determine if two lines are parallel or perpendicular. If the lines are parallel (Figure 3a) then they will have the same gradient, i.e. mAB=mCDmAB=mCD. If the lines are perpendicular (Figure 3b) than we have: -1mAB=mCD-1mAB=mCD

Figure 3
Figure 3 (geom.png)

For example the gradient of the line between the points PP and QQ, with co-ordinates (2;1) and (-2;-2) ((Reference)) is:

Gradient = y 2 - y 1 x 2 - x 1 = - 2 - 1 - 2 - 2 = - 3 - 4 = 3 4 Gradient = y 2 - y 1 x 2 - x 1 = - 2 - 1 - 2 - 2 = - 3 - 4 = 3 4
(1)

The following video provides a summary of the gradient of a line.

Figure 4
Gradient of a line

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