Alg6.2

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

$(12,1),(4,0)\,$

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

$\left(\frac{12+4}{2},\frac{1+0}{2}\right) = \left(8,\frac{1}{2}\right)\,$

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

$d = \sqrt{(12-4)^2+(1-0)^2} = \sqrt{(8)^2+1^2} = \sqrt{64+1} = \sqrt{5\cdot 15} = \sqrt{5\cdot 3\cdot 5} = 5\sqrt{3}\,$

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

$m = \frac{0-1}{4-12} = \frac{-1}{-8} = \frac{1}{8}\,$

The equations of the line are

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

Plug in one known point (say, $(4,0)\,$ ) and the calculated slope.

$0 = \frac{1}{8}\cdot 4 + b\,$

$b = -\frac{4}{8} = -\frac{1}{2}\,$

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

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

Plug in one known point (say, $(12,1)\,$ ) and the calculated slope.

$(y-1) = \frac{1}{8}(x-12)\,$

$y = \frac{1}{8}x - \frac{12}{8} + 1 = \frac{1}{8}x - \frac{4}{8} \,$