The area of a triangle is $\\(frac{1}{2} \times\)$ base $\(\times\)$ height. In our case, the diagram shows that $\(\angle B C D\)$ is a right angle, and we are given the length of $B C$. Therefore, if we can determine the length of $C D$, we can determine the area of triangle $B C D$.

This is a multiple-choice question with one or more correct answers. Let's consider each statement in turn, remembering that for the answer to be correct it must be sufficient by itself-without any of the other state-ments-to enable us to determine the area of $B C D$.

Also remember that you do not need to make each of these calculations when answering the question. The question asks about what information is sufficient to solve for the area of $B C D$.

(A) $\(C G=G D=5 \sqrt{3}\)$

If $C G=G D$, then triangle $C G D$ is isosceles, and so $\(\angle G C D=\angle G D C\)$-let's call that angle $x$.

This means that $\(\angle C G D=180-2 x\)$.

We also then know that $\(\angle C G B=2 x\)$.

And we know that $\(\angle C B G=2 x\)$, since $\(C G=B C =5 \sqrt{3}\)$, making triangle $B C G$ isosceles.

Thus, if $\(\angle C D G=x\)$ and $\(\angle C B G=2 x\)$, then $\(x +2 x=90\)$, since $\(\angle B C D\)$ is 90 and those three angles must sum to 180 .

Since $\(x+2 x=90\)$, then $\(x=30,2 x=60\)$, and triangle $B C D$ is a $\(30: 60: 90\)$ triangle.

Thus, in keeping with the proportions of sides of 30:60:90 triangles, $\(C D=5 \sqrt{3} \times \sqrt{3}=15\)$. Now we know the base and height of the triangle, so we know the area too.
(B) Triangle $B C G$ is an equilateral triangle.

If $B C G$ is an equilateral triangle, then $\(\angle C B G =60\)$, which means that $B C D$ is a $\(30: 60: 90\)$ triangle with sides $\(5 \sqrt{3}\)$ and 15 . These double as our base and height, so we can calculate the area.
(C) $C D=15$

Since $B C D$ is a right triangle, its area is just $\(\frac{1}{2} \times B C \times C D\)$. Since we already have $B C$ and this statement tells us the value of $C D$, we have enough information to calculate the area.
(D) $C G=5 \sqrt{3}$

This by itself does not tell us enough. This statement plus the given tells us that triangle $B C G$ is isosceles, but this information isn't sufficient for determining $C D$ or the area of the triangle.
(E) The length of the hypotenuse of triangle $\(B C D=10 \sqrt{3}\)$

Since $B C D$ is a right triangle, and we began with the value of one of the sides, then the length of the hypotenuse gives us enough information to apply the Pythagorean theorem to determine the value of the other side, $C D$. Then we'll have the values of both the base and the height, with which we can determine the area of the triangle.

Answer choices (A), (B), (C), and (E) are all correct.