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Maths

The straight line

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Lengths, gradients and midpoints

There are several basic facts and equations connected with straight lines that you need to know by heart. Remind yourself of the straight line basics below - you'll need to use these processes in the next section.

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 gradientmbetween two points(x_1 ,y_1 ) and (x_2 ,y_2 ) is given by the formula m = {{y_2 - y_1 } \over {x_2 - x_1 }} 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 = {{5 - 3} \over {1 - 2}} = -2

If a line with gradient m makes an angle aº with the positive direction of the x-axis then m = tan aº

Line with gradient m = tan a

Line with gradient m = tan a

3 = \tan a^\circ so a = 71.6^\circ

Line with gradient m = tan 120

Line with gradient m = tan 120

m = \tan 120^\circ so m = - \sqrt {3}

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