A triangle with angle measures \(30\degree, 60\degree, \text{and } 90 \degree\) has a perimeter of \(9+3\sqrt{3}\). What is the length of the longest side of the triangle?

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The SAT provides us the ratios of the special right triangles:

The perimeter of a 30-60-90 triangle is:

$$ 2x+x+x\sqrt{3} $$ $$ =3x+x\sqrt{3} $$

Comparing this with \(9+3\sqrt{3}\) implies that \(x\) must be 3.

$$ 3(3)+(3)\sqrt{3} = 9+3\sqrt{3} $$

The longest side is therefore \(\boxed{3\sqrt{3}}\).