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Slope of a line

The slope of a line, also called the gradient of a line, is a measure of the steepness of the line. There are two ways to measure the slope of a line – positive and negative.

 

For a given line, when we move from left to right, and the line goes upwards, then we say the positive gradientslope of the line (or gradient of the line) is positive, i.e. it has positive slope.

 

On the other hand, when we move from left to right, and the line goes downwards, then the line isnegative gradient said to have negative slope or negative gradient.

 

 

There is a special type of slope when the line is horizontal, i.e. it is neither going up or down. In thisgradient=0 case, the slope of a line is zero.

 

Formula for slope of a line

formula_for_slope_of_a_line

 

In the diagram, consider the line joining points A (x1, y1) and B (x2, y2). Now drop perpendiculars (shown in dotted lines) from A and B to the x and y axis respectively. This forms a right-angled triangle ABC.

The lines also make an angle theta – the angle of inclination of the line – with the x-axis. So now, let’s consider right-angled DeltaABC where angle BAC = theta (because they are corresponding angles on x-axis and BC).

 

 

Using trigonometry, we know tan theta~=~{opposite side}/{adjacent side}rise_run

=   {y_2~-~y_1}/{x_2~-~x_1}  (for the above diagram)

 

We have also defined earlier the slope of a line to be the steepness of the said line. So the slope of the line will then be equal to tan theta.

In other words, the slope of a line (represented by the letter ‘m’) is given by the formula –

m  =  {y_2~-~y_1}/{x_2~-~x_1}

=  {change in y-axis}/{change in x-axis}

The change in y-axis is called ‘rise‘ since the line rises from a lower point to a higher point. And the change in x-axis is called ‘run‘, since the line runs along the x-axis from left to right. So

m  =  {change in y-axis}/{change in x-axis}

=  {rise}/{run}

=  {y_2~-~y_1}/{x_2~-~x_1}

=  tan theta

where theta is the angle of inclination on a line with x-axis in the positive direction (i.e. anti-clockwise).