# Alg6.3

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

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

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

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

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

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

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

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

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.

$\displaystyle 9 = 0 + b\,$

$\displaystyle b = 9\,$

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

• $\displaystyle y = x + 9\,$

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

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

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

$\displaystyle y = x + 9\,$

• $\displaystyle y = x + 9\,$