If AD is 9 and BD is 3*3^(1/2), what is the area of triangle ABC

Two ways:

First, if you're down with the geometry:
AD is 9 and BD is \(3\sqrt{3}\). That tells us that AB is \(6\sqrt{3}\) and that ABD is a 30-60-90 triangle.
If angle ABD is 60 degrees and angle ABC is 90, we know that angle CBD is 30.
We now have another 30-60-90 triangle and know that since BD is \(3\sqrt{3}\), CD is 3.
AC is 12 and BD is \(3\sqrt{3}\). Area = \(0.5*12*3\sqrt{3}\) = \(18\sqrt{3}\)
Answer choice B.

Or, if you get stuck on geometry questions, just try Ballparking:
\(3\sqrt{3}\) is a touch bigger than 5.
Eyeballing ABD, AD is 9 and BD is a touch bigger than 5. We know we could prove exactly what all three angles are because we already have two sides and could find the third. But let's just ballpark it. One leg is 5 and the other is 9. Sure seems like angle A is right around 30 degrees. That makes angle ABD 60 degrees, which makes CBD 30 degrees. BD is a touch bigger than 5. CD is right around 3.
Okay, let's find the area of ABC. AB is around 12 and BD is a little over 5. Area of ABC is 0.5*12*alittleover5 = a little over 30.
Look at the answer choices. Only one is close.
Answer choice B.