Given (0, 9) and (6, -9), find the midpoint, distance, slope, and equation of the line.

Problem: Given these pairs of points, (0, 9)) and (6, -9), find the midpoint, distance, slope, and equation of the line.

(0,9),(6,-9),

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Solutions:

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

left(frac{0+6}{2},frac{9-9}{2}right) = (3,0),

  • 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{(6)^2+(-9-9)^2} = sqrt{36+18^2} = sqrt{(2cdot 3)^2+(2cdot 3^2)^2} = sqrt{(2cdot 3)^2(1+3^2} = 6sqrt{10},
d = sqrt{(6)^2+(-9-9)^2} = sqrt{36+18^2} = sqrt{(2cdot 3)^2+(2cdot 3^2)^2} = sqrt{(2cdot 3)^2(1+3^2} = 6sqrt{10},

  • To find the slope, use the formula

 m = frac{y_2-y_1}{x_2-x_1},
m = frac{-9-9}{6-0} = -3,
The equations of the line are

Method 1:
 y=mx+b,

  • Plug in one known point (say, (0, 9)) and the calculated slope.

9 = -3cdot 0 + b,
b = 9,

  • Now plug b and m into the line equation:
  • y = -3x + 9,

Method 2:
(y-y_1) = m(x-x_1),

  • Plug in one known point (say, (6, –9) ) and the calculated slope.

(y+9) = -3(x-6),
y = -3x + 18 - 9,

  • y = -3x + 9,

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