distance equation of line Geometry Problems midpoint slope

Given (12, 1) and (4, 0), find the midpoint, distance, slope, and equation of the line.

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Problem: Given these pairs of points, (12, 1) and (4, 0), find the midpoint, distance, slope, and equation of the line.

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

Solutions:

  • 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}\,
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
Method 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:
  • y = \frac{1}{8}x - \frac{1}{2}\,
Method 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} \,
  • y = \frac{1}{8}x - \frac{1}{2}\,

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credit: Todd

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