If 3 points are colinear
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Question

$\text{Determine if the points }A\left(-4,7\right)\text{ , }B\left(-7,13\right)\text{ and }C\left(8,-17\right)\text{ are colinear.}$

Answer

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

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

$\text{The distance between the points }A\left(-4,7\right)\text{ and }B\left(-7,13\right)$

$AB=\sqrt{(13-7)^2+(-7-(-4))^2}$

$= \sqrt{(6)^2+(-3)^2}$

$= \sqrt{36+9}$

$= \sqrt{45}$

$= 6.71$

$\text{The distance between the points }B\left(-7,13\right)\text{ and }C\left(8,-17\right)$

$BC=\sqrt{(-17-13)^2+(8-(-7))^2}$

$= \sqrt{(-30)^2+(15)^2}$

$= \sqrt{900+225}$

$= \sqrt{1125}$

$= 33.54$

$\text{The distance between the points }C\left(8,-17\right)\text{ and }A\left(-4,7\right)$

$CA=\sqrt{(7-(-17))^2+(-4-8)^2}$

$= \sqrt{(24)^2+(-12)^2}$

$= \sqrt{576+144}$

$= \sqrt{720}$

$= 26.83$

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