If 3 points are colinear

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$\text{Determine if the points }A\left(0,2\right)\text{ , }B\left(7,-61\right)\text{ and }C\left(1,-7\right)\text{ are colinear.}$

**Answer**

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

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

$\text{The distance between the points }A\left(0,2\right)\text{ and }B\left(7,-61\right)$

$AB=\sqrt{(-61-2)^2+(7-0)^2}$

$= \sqrt{(-63)^2+(7)^2}$

$= \sqrt{3969+49}$

$= \sqrt{4018}$

$= 63.39$

$\text{The distance between the points }B\left(7,-61\right)\text{ and }C\left(1,-7\right)$

$BC=\sqrt{(-7-(-61))^2+(1-7)^2}$

$= \sqrt{(54)^2+(-6)^2}$

$= \sqrt{2916+36}$

$= \sqrt{2952}$

$= 54.33$

$\text{The distance between the points }C\left(1,-7\right)\text{ and }A\left(0,2\right)$

$CA=\sqrt{(2-(-7))^2+(0-1)^2}$

$= \sqrt{(9)^2+(-1)^2}$

$= \sqrt{81+1}$

$= \sqrt{82}$

$= 9.06$

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