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

(Last Updated On: December 13, 2017)

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{5cdot 15} = sqrt{5cdot 3cdot 5} = 5sqrt{3},
d = sqrt{(12-4)^2+(1-0)^2} = sqrt{(8)^2+1^2} = sqrt{64+1} = sqrt{5cdot 15} = sqrt{5cdot 3cdot 5} = 5sqrt{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},

List of Similar Problems with Complete Solutions

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credit: Todd
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