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# Using Midpoint and Distance Formulas

## Using Midpoint and Distance Formulas 1.14 - Solution

a
Begin by taking note of the coordinates at the endpoints of the distances.
The blue line passes through the coordinates $(\text{-} 7,\text{-} 8)$ and $(\text{-} 3,\text{-} 4).$ Insert these values into the distance formula.
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }$
$d = \sqrt{\left({\color{#0000FF}{\text{-} 3}}-\left({\color{#009600}{\text{-} 7}}\right)\right)^2 + \left({\color{#0000FF}{\text{-} 4}}-\left({\color{#009600}{\text{-} 8}}\right)\right)^2}$
$d=\sqrt{(\text{-}3 + 7)^2+(\text{-}4 + 8)^2}$
$d=\sqrt{4^2+4^2}$
$d=\sqrt{16+16}$
$d=\sqrt{32}$
$d=5.65685\ldots$
$d\approx 5.66$
The blue line's distance is approximately $5.66.$
b
The red line has the endpoints $(3, \text{-} 2)$ and $(6, \text{-}6)$ and we get the length of the line in the same way.
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }$
$d = \sqrt{\left({\color{#0000FF}{6}}-{\color{#009600}{3}}\right)^2 + \left({\color{#0000FF}{\text{-} 6}}-\left({\color{#009600}{\text{-} 2}}\right)\right)^2}$
$d=\sqrt{(6-3)^2+(\text{-}6+2)^2}$
$d=\sqrt{3^2+(\text{-}4)^2}$
$d=\sqrt{9+16}$
$d=\sqrt{25}$
$d=5$
The distance is $5.$
c
Do the same process for the green line, which lies between the points $(\text{-}3, 1)$ and $(7,6).$
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }$
$d = \sqrt{\left({\color{#0000FF}{7}}-\left({\color{#009600}{\text{-} 3}}\right)\right)^2 + \left({\color{#0000FF}{6}}-{\color{#009600}{1}}\right)^2}$
$d=\sqrt{(7+3)^2+(6-1)^2}$
$d=\sqrt{10^2+5^2}$
$d=\sqrt{100+25}$
$d=\sqrt{125}$
$d=11.18033\ldots$
$d\approx11.18$
The distance is $\sim 11.18.$
d
Use the distance formula one last time for the black line, which spans from $(\text{-} 8,2)$ to $(\text{-} 2,6).$
$d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }$
$d = \sqrt{\left({\color{#0000FF}{\text{-} 2}}-\left({\color{#009600}{\text{-} 8}}\right)\right)^2 + \left({\color{#0000FF}{6}}-{\color{#009600}{2}}\right)^2}$
Evaluate right-hand side
$d=\sqrt{(\text{-}2+8)^2+(6-2)^2}$
$d=\sqrt{6^2+4^2}$
$d=\sqrt{36+16}$
$d=\sqrt{52}$
$d=7.21110\ldots$
The black line's length is $\sim 7.21.$