System of Equations

Algebra Level 4

\[ \begin{cases} { a }^{ 2 }+ab+{ b }^{ 2 }=25\\ { b }^{ 2 }+bc+{ c }^{ 2 }=49\\ { c }^{ 2 }+ca+{ a }^{ 2 }=64 \end{cases} \]

Let \(a,b\) and \(c\) be positive real numbers satisfying the system of equations above. Find \((a+b+c)^2\).


This problem is from the Singapore Mathematical Olympiad Junior Round 1 (2011). Full credit is given to the organisers for preparing and deriving the solution to this problem.
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