If 3 points are colinear

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$\text{Determine if the points }A\left(1,0\right)\text{ , }B\left(-4,15\right)\text{ and }C\left(-1,6\right)\text{ are colinear.}$

**Answer**

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

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

$\text{The distance between the points }A\left(1,0\right)\text{ and }B\left(-4,15\right)$

$AB=\sqrt{(15-0)^2+(-4-1)^2}$

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

$= \sqrt{225+25}$

$= \sqrt{250}$

$= 15.81$

$\text{The distance between the points }B\left(-4,15\right)\text{ and }C\left(-1,6\right)$

$BC=\sqrt{(6-15)^2+(-1-(-4))^2}$

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

$= \sqrt{81+9}$

$= \sqrt{90}$

$= 9.49$

$\text{The distance between the points }C\left(-1,6\right)\text{ and }A\left(1,0\right)$

$CA=\sqrt{(0-6)^2+(1-(-1))^2}$

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

$= \sqrt{36+4}$

$= \sqrt{40}$

$= 6.32$

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