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:

\[m = \frac{{5 - 3}}{{1 - 2}} = - 2\]

curriculum-key-fact
  • If a line with gradient \(m\) makes an angle \(\theta\) with the positive direction of the \(x\)-axis then \(m = \tan\theta \)

Example 1

Line with angle theta and point (0, -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

Line with angle 120

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.

\[m=\tan\theta \]

\[m = \tan 120^\circ\]

\[m=-\tan60^\circ\]

\[m = - \sqrt 3\]