If 3 points are colinear

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$\text{Determine if the points }A\left(8,75\right)\text{ , }B\left(6,57\right)\text{ and }C\left(-2,-15\right)\text{ are colinear.}$

**Answer**

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

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

$\text{The distance between the points }A\left(8,75\right)\text{ and }B\left(6,57\right)$

$AB=\sqrt{(57-75)^2+(6-8)^2}$

$= \sqrt{(-18)^2+(-2)^2}$

$= \sqrt{324+4}$

$= \sqrt{328}$

$= 18.11$

$\text{The distance between the points }B\left(6,57\right)\text{ and }C\left(-2,-15\right)$

$BC=\sqrt{(-15-57)^2+(-2-6)^2}$

$= \sqrt{(-72)^2+(-8)^2}$

$= \sqrt{5184+64}$

$= \sqrt{5248}$

$= 72.44$

$\text{The distance between the points }C\left(-2,-15\right)\text{ and }A\left(8,75\right)$

$CA=\sqrt{(75-(-15))^2+(8-(-2))^2}$

$= \sqrt{(90)^2+(10)^2}$

$= \sqrt{8100+100}$

$= \sqrt{8200}$

$= 90.55$

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