Tricky Triangle

Geometry Level 4

\(X\) , \(Y\) and \(Z\) are 3 mutually parallel straight lines in a plane, such that \(Y\) is between \(X\) and \(Z\) and the distance between \(X\) and \(Y\) is \(10\) units and the distance between \(Y\) and \(Z\) is \(6\) units. \(ABC\) is an equilateral triangle with \(A\) on \(X\) , \(B\) on \(Y\) and \(C\) on \(Z\). If the area of \(\triangle ABC\) can be expressed as \(\dfrac{a^2}{\sqrt{b}}\) where \(a\) and \(b\) are coprime positive integers and \(b\) is not divisible by the square of any prime, find the value of \(a+b\).

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