TITLE: Area of Triangle when 2 Sides and No Angle Known
QUESTION [5 upvotes]: It is quite possible this question has no answer -- that is, the area cannot be determined from the information given. It's a question I've created myself as I study for the GRE. No trigonometry is meant to be used, only basic logic about geometry. 
(Please bear with me -- this GRE material is quite different than the formal math I am used to in the areas of analysis, abstract algebra, probability, etc.! -- I want to make sure my logic is sound and that I haven't made any leaps in logic or imposed any extra assumptions that weren't given!)
Please consider the figure below, not necessarily drawn to scale:

My question: can I determine the area of the triangle with the information given?

My idea: Draw a line from $B$ down to a new point $D$ on line $AC$ so that that it creates a right angle with $AC$:

Now, observe triangle $BCD$ above. By construction it is a right triangle, whose hypotenuse is length $5$. Therefore, it must be a "$3$-$4$-$5$" right triangle. 
(Is this step correct or have a made a jump??)
Therefore, I can determine $DC=3$ and $BD=4$.
Finally, I plug in these lengths to compute the area as $A=\frac{1}{2}\cdot 14 \cdot 4 =28$.

REPLY [2 votes]: $a^2+b^2=5^2$ has many solutions, $b=\sqrt{25-a^2}$ for any $a\le 5$.