If 3 points form a right angle triangle

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$\text{Determine if the points }A\left(7,-7\right)\text{ , }B\left(-2,8\right)\text{ and }C \left(2,-10\right)\text{ form a right angle triangle.}$

**Answer**

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

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

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

$\text{The distance between the points }A\left(7,-7\right)\text{ and }B \left(-2,8\right)$

$AB=\sqrt{(8-(-7))^2+(-2-7)^2}$

$= \sqrt{(15)^2+(-9)^2}$

$= \sqrt{225+81}$

$= \sqrt{306}$

$\text{So, }(AB)^2= 306$

$\text{The distance between the points }B \left(-2,8\right)\text{ and }C \left(2,-10\right)$

$BC=\sqrt{(-10-8)^2+(2-(-2))^2}$

$= \sqrt{(-18)^2+(4)^2}$

$= \sqrt{324+16}$

$= \sqrt{340}$

$\text{So, }(BC)^2= 340$

$\text{The distance between the points }C \left(2,-10\right)\text{ and }A \left(7,-7\right)$

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

$= \sqrt{(3)^2+(5)^2}$

$= \sqrt{9+25}$

$= \sqrt{34}$

$\text{So, }(CA)^2=34$

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