Given (0, 9) and (6, -9), find the midpoint, distance, slope, and equation of the line.

Problem: Given these pairs of points, (0, 9)) and (6, -9), find the midpoint, distance, slope, and equation of the line.

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

Solutions:

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

Method 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:
$y = -3x + 9\,$

Method 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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Tags: distance, equation of line, Geometry Problems, midpoint, slope

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Saturday, December 28, 2013