����ý

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:

\(\sqrt {{{(1 - 2)}^2} + {{(5 - 3)}^2}}\)

\(= \sqrt {{{( - 1)}^2} + {{(2)}^2}}\)

\(= \sqrt 5\)

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:

\(m = \frac{{5 - 3}}{{1 - 2}} = - 2\)

\(m = \frac{{{y_2} - {y_1}}}{{{x_2} - {x_1}}} = \frac{5-2}{2-1}=\frac{3}{1}=3\)

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.

\(m=\tan\theta \)

\(m = \tan 120^\circ\)

\(m=-\tan60^\circ\)

\(m = - \sqrt 3\)