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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 P T is the steepest. Line P S is less steep than P T but is steeper than P R , and line P R is steeper than P Q .

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 ( x 1 ; y 1 ) and a point B with co-ordinates ( x 2 ; y 2 ) .

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

Gradient = 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 AB = m CD . If the lines are perpendicular ( [link] b) than we have: - 1 m AB = m CD

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

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

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

Gradient of a line

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Source:  OpenStax, Siyavula textbooks: grade 10 maths [caps]. OpenStax CNX. Aug 03, 2011 Download for free at http://cnx.org/content/col11306/1.4
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