If 3 points are colinear

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$\text{Determine if the points }A\left(4,-17\right)\text{ , }B\left(-1,-2\right)\text{ and }C\left(8,-29\right)\text{ are colinear.}$

**Answer**

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

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

$\text{The distance between the points }A\left(4,-17\right)\text{ and }B\left(-1,-2\right)$

$AB=\sqrt{(-2-(-17))^2+(-1-4)^2}$

$= \sqrt{(15)^2+(-5)^2}$

$= \sqrt{225+25}$

$= \sqrt{250}$

$= 15.81$

$\text{The distance between the points }B\left(-1,-2\right)\text{ and }C\left(8,-29\right)$

$BC=\sqrt{(-29-(-2))^2+(8-(-1))^2}$

$= \sqrt{(-27)^2+(9)^2}$

$= \sqrt{729+81}$

$= \sqrt{810}$

$= 28.46$

$\text{The distance between the points }C\left(8,-29\right)\text{ and }A\left(4,-17\right)$

$CA=\sqrt{(-17-(-29))^2+(4-8)^2}$

$= \sqrt{(12)^2+(-4)^2}$

$= \sqrt{144+16}$

$= \sqrt{160}$

$= 12.65$

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