If 3 points form a right angle triangle

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$\text{Determine if the points }A\left(9,-2\right)\text{ , }B\left(4,3\right)\text{ and }C \left(-6,-17\right)\text{ form a right angle triangle.}$

**Answer**

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

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

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

$\text{The distance between the points }A\left(9,-2\right)\text{ and }B \left(4,3\right)$

$AB=\sqrt{(3-(-2))^2+(4-9)^2}$

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

$= \sqrt{25+25}$

$= \sqrt{50}$

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

$\text{The distance between the points }B \left(4,3\right)\text{ and }C \left(-6,-17\right)$

$BC=\sqrt{(-17-3)^2+(-6-4)^2}$

$= \sqrt{(-20)^2+(-10)^2}$

$= \sqrt{400+100}$

$= \sqrt{500}$

$\text{So, }(BC)^2= 500$

$\text{The distance between the points }C \left(-6,-17\right)\text{ and }A \left(9,-2\right)$

$CA=\sqrt{(-2-(-17))^2+(9-(-6))^2}$

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

$= \sqrt{225+225}$

$= \sqrt{450}$

$\text{So, }(CA)^2=450$

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