Given (-2, 10) and (4, 9), find the midpoint, distance, slope, and equation of the line.

Problem: Given these pairs of points, (-2, 10) and (4, 9), find the midpoint, distance, slope, and equation of the line.

$(-2,10),(4,9)\,$

Solutions:

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

$\left(\frac{-2+4}{2},\frac{10+9}{2}\right) = \left(\frac{2}{2},\frac{19}{2}\right) = \left(1,\frac{19}{2}\right)\,$

To find the (always zero or positive) distance, use the formula

$d = +\sqrt{(x_1-x_2)^2 + (y_1-y_2)^2}\,$

$d = \sqrt{(-2-4)^2+(10-9)^2} = \sqrt{(-6)^2+1^2} = \sqrt{36+1} = \sqrt{37}\,$

To find the slope, use the formula

$m = \frac{y_2-y_1}{x_2-x_1}\,$

$m = \frac{9-10}{4-(-2)} = \frac{-1}{6}\,$

The equation of the line are

Method 1:

$y=mx+b\,$

Plug in one known point (say, (4, 9)) and the calculated slope.

$9 = \frac{-1}{6}\cdot 4 + b\,$

$b = 9+\frac{2}{3} = \frac{29}{3}\,$

Now plug b and m into the line equation:

$y = \frac{-1}{6}x + \frac{29}{3}\,$

Method 2:

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

Plug in one known point (say, (-2, 10)) and the calculated slope.

$(y-10) = \frac{-1}{6}(x-(-2))\,$

$y = \frac{-1}{6}x - \frac{2}{6} + 10 = \frac{-1}{6}x + \frac{58}{6} \,$

$y = \frac{-1}{6}x + \frac{29}{3}\,$

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

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Saturday, December 28, 2013