If 3 points form a right angle triangle

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$\text{Determine if the points }A\left(-2,6\right)\text{ , }B\left(-11,12\right)\text{ and }C \left(2,12\right)\text{ form a right angle triangle.}$

**Answer**

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

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

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

$\text{The distance between the points }A\left(-2,6\right)\text{ and }B \left(-11,12\right)$

$AB=\sqrt{(12-6)^2+(-11-(-2))^2}$

$= \sqrt{(6)^2+(-9)^2}$

$= \sqrt{36+81}$

$= \sqrt{117}$

$\text{So, }(AB)^2= 117$

$\text{The distance between the points }B \left(-11,12\right)\text{ and }C \left(2,12\right)$

$BC=\sqrt{(12-12)^2+(2-(-11))^2}$

$= \sqrt{(0)^2+(13)^2}$

$= \sqrt{0+169}$

$= \sqrt{169}$

$\text{So, }(BC)^2= 169$

$\text{The distance between the points }C \left(2,12\right)\text{ and }A \left(-2,6\right)$

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

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

$= \sqrt{36+16}$

$= \sqrt{52}$

$\text{So, }(CA)^2=52$

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