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 xaxis.

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