If 3 points form a right angle triangle

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$\text{Determine if the points }A\left(-6,7\right)\text{ , }B\left(-12,19\right)\text{ and }C \left(-10,5\right)\text{ form a right angle triangle.}$

**Answer**

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

$\text{is the sum of the squares of the other two distances, the 3 points will form a right}$

$\text{angle triangle, otherwise not.}$

$\text{The distance between the points }A\left(-6,7\right)\text{ and }B \left(-12,19\right)$

$AB=\sqrt{(19-7)^2+(-12-(-6))^2}$

$= \sqrt{(12)^2+(-6)^2}$

$= \sqrt{144+36}$

$= \sqrt{180}$

$\text{So, }(AB)^2= 180$

$\text{The distance between the points }B \left(-12,19\right)\text{ and }C \left(-10,5\right)$

$BC=\sqrt{(5-19)^2+(-10-(-12))^2}$

$= \sqrt{(-14)^2+(2)^2}$

$= \sqrt{196+4}$

$= \sqrt{200}$

$\text{So, }(BC)^2= 200$

$\text{The distance between the points }C \left(-10,5\right)\text{ and }A \left(-6,7\right)$

$CA=\sqrt{(7-5)^2+(-6-(-10))^2}$

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

$= \sqrt{4+16}$

$= \sqrt{20}$

$\text{So, }(CA)^2=20$

$\text{Since, }(BC)^2 = (AB)^2 + (CA)^2 \text{ , the 3 points form a right angle triangle.}$