If 3 points are colinear

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$\text{Determine if the points }A\left(-6,-22\right)\text{ , }B\left(-4,-16\right)\text{ and }C\left(4,8\right)\text{ are colinear.}$

**Answer**

$\text{We will find the distance between each pair of points. If one distance is the sum}$

$\text{of the other two distances, the 3 points are colinear otherwise not colinear.}$

$\text{The distance between the points }A\left(-6,-22\right)\text{ and }B\left(-4,-16\right)$

$AB=\sqrt{(-16-(-22))^2+(-4-(-6))^2}$

$= \sqrt{(6)^2+(2)^2}$

$= \sqrt{36+4}$

$= \sqrt{40}$

$= 6.32$

$\text{The distance between the points }B\left(-4,-16\right)\text{ and }C\left(4,8\right)$

$BC=\sqrt{(8-(-16))^2+(4-(-4))^2}$

$= \sqrt{(24)^2+(8)^2}$

$= \sqrt{576+64}$

$= \sqrt{640}$

$= 25.30$

$\text{The distance between the points }C\left(4,8\right)\text{ and }A\left(-6,-22\right)$

$CA=\sqrt{(-22-8)^2+(-6-4)^2}$

$= \sqrt{(-30)^2+(-10)^2}$

$= \sqrt{900+100}$

$= \sqrt{1000}$

$= 31.62$

$\text{Since }CA = AB + BC\text{, the 3 points are colinear.}$