What are some ways to check if a the information given is enough to solve a problem related to euclidean geometry? To know if a the data given produces a unique answer is something important because if you know the data is insufficient to yield a unique answer you can stop looking for one. 

Example: $\triangle ABC$ is right angled at $B$. A perpendicular $BD$ is dropped on to $AC$. Given that $|AD|=4cm$ and $|DC|=12cm$. Find the length of $BC$.

In this problem first mentally "drawing" a line $AC$ of length $16cm$ in our head and drawing a perpendicular at the required point $D$ and then considering various points on the perpendicular helps us "know" that there is a unique point where the angle is $90 ^\circ$. So, we know the data is sufficient.
Are there any other techniques which help us check if the data is sufficient?
 A: As you said, 'mental drawing' is one technique: this can actually be formalized through geometric constructions. Given some values (angles, lengths, relationships between angles/lengths, etc.) can you construct a valid diagram? Sometimes it is helpful to consider compass-and-ruler only constructions, sometimes you can allow the construction of conics, and so on, but essentially: method 1 -- construct it.
There is also a 'brute-force' approach using coordinate geometry. This is often inelegant, but it can give some insight on the problem. Here, you try to compute every single angle and length in a given diagram with elementary geometry and trigonometry, using placeholder variables ($x$, $\theta$, etc.) when you don't know some of the values. At the end of the process (when you've computed everything that it is possible to compute) if you're left with free variables, then (a.) you need more information or (b.) the property in question holds regardless of the value of that variable. The second conclusion sounds weak, but it can be a very helpful insight. So method 2 -- calculate it.
(Note: Above, I said you can use elementary geometry and trigonometry. I think these tools are sufficiently powerful to compute everything in a diagram, but it may not be so.)
A third technique which may require a little more effort is to reduce the problem to a known, solved problem. If, by adding constraints or changing them slightly, you can reduce the problem to a previous one, then you can sometimes prove that reverting those changes makes the problem unsolvable. The method used to solve the problem may also be generalizable. Though not the most useful method, method 3 -- reduce it.
