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. AD 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 

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. 

With the RedCrab calculater use the function Grad to calculate the gradient. 
Grad(Point(1,1), Point(2,5)) = 4



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 xaxis 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.


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