Given (4, 13) and (0, 9), find the midpoint, distance, slope, and equation of the line.

Problem: Given these pairs of points, (4, 13) and (0, 9), find the midpoint, distance, slope, and equation of the line.

$(4,13),(09,0)\,$

Solutions:

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

Method 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:

$y = x + 9\,$

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

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Saturday, December 28, 2013