Let \(\color{brown}{\Delta ABC}\) be a triangle with side \(\color{red}{AB = 12,} ~\color{blue}{BC=4\sqrt{5}}~ \& ~ \color{orange}{AC = 8\sqrt{2}}\).
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Also let \(\color{grey}{P,~Q~\& ~R}\) be the points on side \(\color{indigo}{AB,~BC~ \& ~ CA} \) respectively such that perimeter of \(\color{green}{\Delta PQR}\) is minimum.

Find the inradius of \(\color{green}{\Delta PQR}\).

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