Mirror, Mirror

Geometry Level 5

The minimum possible perimeter of a triangle with one vertex at \((3,9)\), one anywhere on the \(y\)-axis and one anywhere on the line \(y = x\) is \(a\sqrt{b}\), where \(a\) and \(b\) are positive coprime integers and \(b\) is square free. Find \(a + b.\)

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