If 3 points are colinear

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$\text{Determine if the points }A\left(7,-41\right)\text{ , }B\left(-1,7\right)\text{ and }C\left(-6,37\right)\text{ are colinear.}$

**Answer**

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

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

$\text{The distance between the points }A\left(7,-41\right)\text{ and }B\left(-1,7\right)$

$AB=\sqrt{(7-(-41))^2+(-1-7)^2}$

$= \sqrt{(48)^2+(-8)^2}$

$= \sqrt{2304+64}$

$= \sqrt{2368}$

$= 48.66$

$\text{The distance between the points }B\left(-1,7\right)\text{ and }C\left(-6,37\right)$

$BC=\sqrt{(37-7)^2+(-6-(-1))^2}$

$= \sqrt{(30)^2+(-5)^2}$

$= \sqrt{900+25}$

$= \sqrt{925}$

$= 30.41$

$\text{The distance between the points }C\left(-6,37\right)\text{ and }A\left(7,-41\right)$

$CA=\sqrt{(-41-37)^2+(7-(-6))^2}$

$= \sqrt{(-78)^2+(13)^2}$

$= \sqrt{6084+169}$

$= \sqrt{6253}$

$= 79.08$

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