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

Saturday, December 28, 2013

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

Posted by Unknown at 10:33 PM

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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