# Alg6.5

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

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

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

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

To find the (always zero or positive) distance, use the formula $\displaystyle d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,$

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

To find the slope, use the formula $\displaystyle m = \frac{y_2-y_1}{x_2-x_1}\,$

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

The equations of the line are

Form 1: $\displaystyle y=mx+b\,$

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

$\displaystyle 9 = -3\cdot 0 + b\,$

$\displaystyle b = 9\,$

Now plug $\displaystyle b$ and $\displaystyle m$ into the line equation:

• $\displaystyle y = -3x + 9\,$

Form 2: $\displaystyle (y-y_1) = m(x-x_1)\,$

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

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

$\displaystyle y = -3x + 18 - 9\,$

• $\displaystyle y = -3x + 9\,$