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Question$\text{Determine if the points }A\left(-2,-9\right)\text{ , }B\left(-9,-37\right)\text{ and }C\left(-10,-41\right)\text{ are colinear.}$
Answer$\text{We will find the distance between each pair of points. If one distance is the sum}$
$\text{of the other two distances, the 3 points are colinear otherwise not colinear.}$
$\text{The distance between the points }A\left(-2,-9\right)\text{ and }B\left(-9,-37\right)$
$AB=\sqrt{(-37-(-9))^2+(-9-(-2))^2}$
$= \sqrt{(-28)^2+(-7)^2}$
$= \sqrt{784+49}$
$= \sqrt{833}$
$= 28.86$
$\text{The distance between the points }B\left(-9,-37\right)\text{ and }C\left(-10,-41\right)$
$BC=\sqrt{(-41-(-37))^2+(-10-(-9))^2}$
$= \sqrt{(-4)^2+(-1)^2}$
$= \sqrt{16+1}$
$= \sqrt{17}$
$= 4.12$
$\text{The distance between the points }C\left(-10,-41\right)\text{ and }A\left(-2,-9\right)$
$CA=\sqrt{(-9-(-41))^2+(-2-(-10))^2}$
$= \sqrt{(32)^2+(8)^2}$
$= \sqrt{1024+64}$
$= \sqrt{1088}$
$= 32.98$
$\text{Since }CA = AB + BC\text{, the 3 points are colinear.}$