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 $\boxed{3\sqrt{3}}$.