Triangles

[WilfordL612]

This is from Manhattan CAT 4

If AD is 6√3, and ADC is a right angle, what is the area of triangular region ABC?

(1) Angle ABD = 60°

(2) AC = 12

https://cdn2.manhattanprep.com/images/g ... gleabc.jpg

I answered D and the correct answer is C.

My reasoning:

(1) If angle ABD = 60 degrees then this is a 30-60-90 triangle. Therefore the ratio is 6 : 6√3 : 12. Since AD bisects the triangle I naturally assumed that the total base (BC) would be 12.

(2) If AC = 12 and AD = 6√3, I can use Pyth. Theorem and get b = 6. Since AD bisects the triangle I naturally assumed that the total base (BC) would be 12.

Now in the explanations it just mentions that I only know one side of the split triangle and calls each statement alone insufficient. So my question is what mistake am I making? The question stem doesn't say AD bisects the triangle but wouldn't I naturally assume this because it gives me a right angle? What am I missing here?

Any help is appreciated.

[Sage Pearce-Higgins]

Thanks for expressing your reasoning so clearly. My only disagreement is your line

Since AD bisects the triangle

There's nothing in the problem that tells you this. It's really important not to be tricked by the diagrams in Geometry problems - they are often not drawn to scale (and they only sometimes state 'not drawn to scale'). If you remove this assumption then DC could be any length, and the triangle would have many possible areas. This is why statement (1) alone isn't sufficient.