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Using Midpoint and Distance Formulas

Using Midpoint and Distance Formulas 1.14 - Solution

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a
Begin by taking note of the coordinates at the endpoints of the distances.
The blue line passes through the coordinates (-7,-8)(\text{-} 7,\text{-} 8) and (-3,-4).(\text{-} 3,\text{-} 4). Insert these values into the distance formula.
d=(x2x1)2+(y2y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }
d=(-3(-7))2+(-4(-8))2d = \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=(-3+7)2+(-4+8)2d=\sqrt{(\text{-}3 + 7)^2+(\text{-}4 + 8)^2}
d=42+42d=\sqrt{4^2+4^2}
d=16+16d=\sqrt{16+16}
d=32d=\sqrt{32}
d=5.65685d=5.65685\ldots
d5.66d\approx 5.66
The blue line's distance is approximately 5.66.5.66.
b
The red line has the endpoints (3,-2)(3, \text{-} 2) and (6,-6)(6, \text{-}6) and we get the length of the line in the same way.
d=(x2x1)2+(y2y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }
d=(63)2+(-6(-2))2d = \sqrt{\left({\color{#0000FF}{6}}-{\color{#009600}{3}}\right)^2 + \left({\color{#0000FF}{\text{-} 6}}-\left({\color{#009600}{\text{-} 2}}\right)\right)^2}
d=(63)2+(-6+2)2d=\sqrt{(6-3)^2+(\text{-}6+2)^2}
d=32+(-4)2d=\sqrt{3^2+(\text{-}4)^2}
d=9+16d=\sqrt{9+16}
d=25d=\sqrt{25}
d=5d=5
The distance is 5.5.
c
Do the same process for the green line, which lies between the points (-3,1)(\text{-}3, 1) and (7,6).(7,6).
d=(x2x1)2+(y2y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }
d=(7(-3))2+(61)2d = \sqrt{\left({\color{#0000FF}{7}}-\left({\color{#009600}{\text{-} 3}}\right)\right)^2 + \left({\color{#0000FF}{6}}-{\color{#009600}{1}}\right)^2}
d=(7+3)2+(61)2d=\sqrt{(7+3)^2+(6-1)^2}
d=102+52d=\sqrt{10^2+5^2}
d=100+25d=\sqrt{100+25}
d=125d=\sqrt{125}
d=11.18033d=11.18033\ldots
d11.18d\approx11.18
The distance is 11.18.\sim 11.18.
d
Use the distance formula one last time for the black line, which spans from (-8,2)(\text{-} 8,2) to (-2,6).(\text{-} 2,6).
d=(x2x1)2+(y2y1)2d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 }
d=(-2(-8))2+(62)2d = \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=(-2+8)2+(62)2d=\sqrt{(\text{-}2+8)^2+(6-2)^2}
d=62+42d=\sqrt{6^2+4^2}
d=36+16d=\sqrt{36+16}
d=52d=\sqrt{52}
d=7.21110d=7.21110\ldots
The black line's length is 7.21.\sim 7.21.