Can you help me find the area of the gray triangle in the given figure. I'm having a hard time finding the base value of the triangle, I've managed to find the sides for the big triangle but not for the greyed area. Thanks in advance to those that will help.
As mentioned in the comments, there simply isn't enough information. All you know about the grayed triangle is that it is a right triangle with one of its legs having length $4,$ and the other having length somewhere between $0$ and $4\sqrt3.$ With only this information, the best we can conclude is that the grayed triangle can have any area between $0$ and $8\sqrt3.$ (And I do mean any such area. All we have to do is position $C$ appropriately.)
Is there perhaps some other information that you left out because you weren't sure how it was relevant, or because you missed it on your first read-through?