This question was uploaded on 27/01/22 on social media accounts.
Solution:
By applying cosine rule opposite to side A:
\[\color{black} {B^2+C^2-A^2=2BC\cos{x}}\]
By applying cosine rule opposite to angle X:
\[\color{black} {B^2+C^2-3^2=2BC\cos{(180-x)}}\]
\[\Rightarrow \color{black} {3^2-(B^2+C^2)=2BC\cos{x}}\]
From both above equations:
\[\color{black} {B^2+C^2-A^2=3^2-(B^2+C^2)}\]
\[\Rightarrow \color{Blue} {2(B^2+C^2)-A^2=3^2}\]
Similarly:
\[\Rightarrow \color{Red} {2(A^2+C^2)-B^2=4^2}\]
\[\Rightarrow \color{Green} {2(A^2+B^2)-C^2=5^2}\]
By adding the above three equations:
\[\color{Purple} {4(A^2+B^2+C^2)-(A^2+B^2+C^2)=3^2+4^2+5^2}\]
\[\Rightarrow \color{Purple} {A_{Blue}=A^2+B^2+C^2=\frac{50}3}\]
