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Question$\text{Determine if the points }A\left(-8,-44\right)\text{ , }B\left(-3,-14\right)\text{ and }C\left(9,58\right)\text{ are colinear.}$
Answer$\text{We will find the distance between each pair of points. If one distance is the sum}$
$\text{of the other two distances, the 3 points are colinear otherwise not colinear.}$
$\text{The distance between the points }A\left(-8,-44\right)\text{ and }B\left(-3,-14\right)$
$AB=\sqrt{(-14-(-44))^2+(-3-(-8))^2}$
$= \sqrt{(30)^2+(5)^2}$
$= \sqrt{900+25}$
$= \sqrt{925}$
$= 30.41$
$\text{The distance between the points }B\left(-3,-14\right)\text{ and }C\left(9,58\right)$
$BC=\sqrt{(58-(-14))^2+(9-(-3))^2}$
$= \sqrt{(72)^2+(12)^2}$
$= \sqrt{5184+144}$
$= \sqrt{5328}$
$= 72.99$
$\text{The distance between the points }C\left(9,58\right)\text{ and }A\left(-8,-44\right)$
$CA=\sqrt{(-44-58)^2+(-8-9)^2}$
$= \sqrt{(-102)^2+(-17)^2}$
$= \sqrt{10404+289}$
$= \sqrt{10693}$
$= 103.41$
$\text{Since }CA = AB + BC\text{, the 3 points are colinear.}$