If 3 points form a right angle triangle

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$\text{Determine if the points }A\left(-5,-3\right)\text{ , }B\left(4,3\right)\text{ and }C \left(-7,0\right)\text{ form a right angle triangle.}$

**Answer**

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

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

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

$\text{The distance between the points }A\left(-5,-3\right)\text{ and }B \left(4,3\right)$

$AB=\sqrt{(3-(-3))^2+(4-(-5))^2}$

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

$= \sqrt{36+81}$

$= \sqrt{117}$

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

$\text{The distance between the points }B \left(4,3\right)\text{ and }C \left(-7,0\right)$

$BC=\sqrt{(0-3)^2+(-7-4)^2}$

$= \sqrt{(-3)^2+(-11)^2}$

$= \sqrt{9+121}$

$= \sqrt{130}$

$\text{So, }(BC)^2= 130$

$\text{The distance between the points }C \left(-7,0\right)\text{ and }A \left(-5,-3\right)$

$CA=\sqrt{(-3-0)^2+(-5-(-7))^2}$

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

$= \sqrt{9+4}$

$= \sqrt{13}$

$\text{So, }(CA)^2=13$

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