If 3 points form a right angle triangle

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$\text{Determine if the points }A\left(8,5\right)\text{ , }B\left(6,4\right)\text{ and }C \left(5,11\right)\text{ form a right angle triangle.}$

**Answer**

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

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

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

$\text{The distance between the points }A\left(8,5\right)\text{ and }B \left(6,4\right)$

$AB=\sqrt{(4-5)^2+(6-8)^2}$

$= \sqrt{(-1)^2+(-2)^2}$

$= \sqrt{1+4}$

$= \sqrt{5}$

$\text{So, }(AB)^2= 5$

$\text{The distance between the points }B \left(6,4\right)\text{ and }C \left(5,11\right)$

$BC=\sqrt{(11-4)^2+(5-6)^2}$

$= \sqrt{(7)^2+(-1)^2}$

$= \sqrt{49+1}$

$= \sqrt{50}$

$\text{So, }(BC)^2= 50$

$\text{The distance between the points }C \left(5,11\right)\text{ and }A \left(8,5\right)$

$CA=\sqrt{(5-11)^2+(8-5)^2}$

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

$= \sqrt{36+9}$

$= \sqrt{45}$

$\text{So, }(CA)^2=45$

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