Alg6.3

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

(4,13),(0,9)\,

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

\left({\frac  {4+0}{2}},{\frac  {13+9}{2}}\right)=(2,11),

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  {(4-0)^{2}+(13-9)^{2}}}={\sqrt  {(4)^{2}+4^{2}}}={\sqrt  {16+16}}={\sqrt  {16*2}}={\sqrt  {4*4*2}}=4{\sqrt  {2}}\,

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

m={\frac  {9-13}{0-4}}={\frac  {-4}{-4}}=1,

The equations of the line are

Form 1: y=mx+b\,

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

9=0+b\,

b=9\,

Now plug b and m into the line equation:

  • y=x+9\,

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

Plug in one known point (say, (4,13)\,) and the calculated slope.

(y-13)=(x-4)\,

y=x+9\,

  • y=x+9\,

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