If 3 points are colinear

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$\text{Determine if the points }A\left(-6,28\right)\text{ , }B\left(5,-16\right)\text{ and }C\left(7,-24\right)\text{ are colinear.}$

**Answer**

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

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

$\text{The distance between the points }A\left(-6,28\right)\text{ and }B\left(5,-16\right)$

$AB=\sqrt{(-16-28)^2+(5-(-6))^2}$

$= \sqrt{(-44)^2+(11)^2}$

$= \sqrt{1936+121}$

$= \sqrt{2057}$

$= 45.35$

$\text{The distance between the points }B\left(5,-16\right)\text{ and }C\left(7,-24\right)$

$BC=\sqrt{(-24-(-16))^2+(7-5)^2}$

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

$= \sqrt{64+4}$

$= \sqrt{68}$

$= 8.25$

$\text{The distance between the points }C\left(7,-24\right)\text{ and }A\left(-6,28\right)$

$CA=\sqrt{(28-(-24))^2+(-6-7)^2}$

$= \sqrt{(52)^2+(-13)^2}$

$= \sqrt{2704+169}$

$= \sqrt{2873}$

$= 53.60$

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