# 8.1 Calculation of the gradient line

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## Analytical geometry; calculation of the gradient line

The gradient of a line describes how steep the line is. In the figure, line $PT$ is the steepest. Line $PS$ is less steep than $PT$ but is steeper than $PR$ , and line $PR$ is steeper than $PQ$ .

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 $A$ with co-ordinates $\left({x}_{1};{y}_{1}\right)$ and a point $B$ with co-ordinates $\left({x}_{2};{y}_{2}\right)$ .

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

$\mathrm{Gradient}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}$

We can use the gradient of a line to determine if two lines are parallel or perpendicular. If the lines are parallel ( [link] a) then they will have the same gradient, i.e. ${m}_{\mathrm{AB}}={m}_{\mathrm{CD}}$ . If the lines are perpendicular ( [link] b) than we have: $-\frac{1}{{m}_{\mathrm{AB}}}={m}_{\mathrm{CD}}$

For example the gradient of the line between the points $P$ and $Q$ , with co-ordinates (2;1) and (-2;-2) ( [link] ) is:

$\begin{array}{ccc}\hfill \mathrm{Gradient}& =& \frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\hfill \\ & =& \frac{-2-1}{-2-2}\hfill \\ & =& \frac{-3}{-4}\hfill \\ & =& \frac{3}{4}\hfill \end{array}$

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

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