If 3 points are colinear

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$\text{Determine if the points }A\left(-9,-8\right)\text{ , }B\left(4,5\right)\text{ and }C\left(-4,-3\right)\text{ are colinear.}$

**Answer**

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

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

$\text{The distance between the points }A\left(-9,-8\right)\text{ and }B\left(4,5\right)$

$AB=\sqrt{(5-(-8))^2+(4-(-9))^2}$

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

$= \sqrt{169+169}$

$= \sqrt{338}$

$= 18.38$

$\text{The distance between the points }B\left(4,5\right)\text{ and }C\left(-4,-3\right)$

$BC=\sqrt{(-3-5)^2+(-4-4)^2}$

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

$= \sqrt{64+64}$

$= \sqrt{128}$

$= 11.31$

$\text{The distance between the points }C\left(-4,-3\right)\text{ and }A\left(-9,-8\right)$

$CA=\sqrt{(-8-(-3))^2+(-9-(-4))^2}$

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

$= \sqrt{25+25}$

$= \sqrt{50}$

$= 7.07$

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