If 3 points are colinear
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Question

$\text{Determine if the points }A\left(9,-86\right)\text{ , }B\left(-9,76\right)\text{ and }C\left(1,-14\right)\text{ are colinear.}$

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

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

$\text{The distance between the points }A\left(9,-86\right)\text{ and }B\left(-9,76\right)$

$AB=\sqrt{(76-(-86))^2+(-9-9)^2}$

$= \sqrt{(162)^2+(-18)^2}$

$= \sqrt{26244+324}$

$= \sqrt{26568}$

$= 163.00$

$\text{The distance between the points }B\left(-9,76\right)\text{ and }C\left(1,-14\right)$

$BC=\sqrt{(-14-76)^2+(1-(-9))^2}$

$= \sqrt{(-90)^2+(10)^2}$

$= \sqrt{8100+100}$

$= \sqrt{8200}$

$= 90.55$

$\text{The distance between the points }C\left(1,-14\right)\text{ and }A\left(9,-86\right)$

$CA=\sqrt{(-86-(-14))^2+(9-1)^2}$

$= \sqrt{(-72)^2+(8)^2}$

$= \sqrt{5184+64}$

$= \sqrt{5248}$

$= 72.44$

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