Gradient of lines

Description for the calculation of the gradient of lines

The Gradient of a Line

The gradient of a line segment is a measure of how steep the line is. A line with a large gradient will be steep; a line with a small gradient will be relatively shallow; and a line with zero gradient will be horizontal.

The next figure shows three line segments. The line segment \(AD\) is steeper than the line segment \(AC\). \(AC\) is steeper than \(AB\), which is horizontal. The steepness is quantify mathematically by measuring the relative changes in \(x\) and \(y\) as we move from the beginning to the end of the line segment.

On the segment\(AD\), \(y\) changes from \(1\) to \(5\), as \(x\) changes from \(1\) to \(2\). So the change in \(y\) is \(4\), and the change in \(x\) is \(1\). The relative change, the gradient of the line segment is

\(\displaystyle \frac{Differenz\; Y}{Differenz\; X} =\frac{5-1}{2-1}=\frac{4}{1}=4\)

Lines which are steeper have a larger gradient than lines which are less steep. The gradient of horizontal lines is zero.

The gradient of a line is equal to the tangent of the angle that the line makes with the horizontal. This is also the tangent of the angle the line makes with the x axis.

Parallel Lines

Apply the knowledge of gradients to the case of parallel lines. If line \(1\) and line \(2\) are two parallel lines, then the angles \(θ1\) and \(θ2\)that they make with the x-axis are corresponding angles, and so must be equal. Therefore parallel lines must have the same gradient.

The next figure shows an example, generated with RedCrab calculator. It shows two parallel lines with the gradient \(m = 1.333\).