Basics of straight lines
There are several basic facts and equations connected with straight lines that you need to know by heart.
Distance formula
The distance between two points, \(({x_1},{y_1})\) and \(({x_2},{y_2})\) is given by the formula:
\(\sqrt {{{({x_2} - {x_1})}^2} + {{({y_2} - {y_1})}^2}}\)
So the distance between \((2,3)\) and \((1,5)\) is:
\(\sqrt {{{(1 - 2)}^2} + {{(5 - 3)}^2}}\)
\(= \sqrt {{{( - 1)}^2} + {{(2)}^2}}\)
\(= \sqrt 5\)
Gradient
The gradient m between two points \(({x_1},{y_1})\) and \(({x_2},{y_2})\) is given by the formula:
\(m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}}\)
This only applies where \({x_2} \ne {x_1}\). If \({x_2} = {x_1}\) then the gradient is undefined.
The gradient between \((2,3)\) and \((1,5)\) is:
\(m = \frac{{5 - 3}}{{1 - 2}} = - 2\)
Example 1
We can calculate the gradient of the line above by selecting two coordinate points that the straight line passes through.
(1,2) and (2,5)
\(m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \frac{5-2}{2-1}=\frac{3}{1}=3\)
Now that we know the gradient, we can calculate the angle that the straight line makes with the positive direction of the \(x\)axis.
\(m=\tan\theta\)
\(3=\tan\theta\)
\(\theta =\tan^{-1}(3)\)
\(\theta =71.6^\circ\)
Similarly, we can calculate the gradient of the straight line if we know the angle the line makes with the positive direction of the \(x\) axis.
Example 2
This time you will notice, the line has a negative gradient, so you will need to use your knowledge of quartiles to calculate this gradient.
\(m=\tan\theta \)
\(m = \tan 120^\circ\)
\(m=-\tan60^\circ\)
\(m = - \sqrt 3\)
