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The straight line


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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