Alg6.5

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For these pairs of points, find the midpoint, distance, slope, and equation of the line.

(0,9),(6,-9)\,

To find the midpoint, average the x coordinates and y coordinates. The midpoint is

\left({\frac  {0+6}{2}},{\frac  {9-9}{2}}\right)=(3,0)\,

To find the (always zero or positive) distance, use the formula d=+{\sqrt  {(x_{1}-x_{2})^{2}+(y_{1}-y_{2})^{2}}}\,

d={\sqrt  {(6)^{2}+(-9-9)^{2}}}={\sqrt  {36+18^{2}}}={\sqrt  {(2\cdot 3)^{2}+(2\cdot 3^{2})^{2}}}={\sqrt  {(2\cdot 3)^{2}(1+3^{2}}}=6{\sqrt  {10}}\,

To find the slope, use the formula m={\frac  {y_{2}-y_{1}}{x_{2}-x_{1}}}\,

m={\frac  {-9-9}{6-0}}=-3\,

The equations of the line are

Form 1: y=mx+b\,

Plug in one known point (say, (0,9)\,) and the calculated slope.

9=-3\cdot 0+b\,

b=9\,

Now plug b and m into the line equation:

  • y=-3x+9\,

Form 2: (y-y_{1})=m(x-x_{1})\,

Plug in one known point (say, (6,-9)\,) and the calculated slope.

(y+9)=-3(x-6)\,

y=-3x+18-9\,

  • y=-3x+9\,

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