Is this trigonometry question possible?

Question

OK first of all, I'm not looking for someone to give me the answer to my homework. I just want to know if this is possible or our maths lecturer has made a mistake.

The question is find the length of $\text{BD}$. I've worked out the length of $\text{CB}$ is $11.97\text{cm}$, but that's as far as I've gotten. Am I right in thinking to determine $\text{BD}$ I need another angle or length in the triangle on the right. I also realised this is not to scale but this is the way it was given to us.

Again, I'm not looking for the answer, I just want to know if it's possible, thank you.

Answer 1

(I am assuming $\angle(BDE)=90^o$. If this isn't the case, then $BD$ is not fully defined.)

It seems you already computed, $$CB = 12\sin(86^o)$$

Notice that, $$BD = |CD - CB|$$

I put absolute values because the difference I wrote is based on how you drew the figure, as though $CD > CB$. In reality $CD < CB$ so it looks more like this:

It is possible to compute $CD$ and thus possible to compute $BD$.

Hint: make use of $\cos$.

Answer 2

Are you told that $\angle CDE$ and $\angle CBA$ are right angles? If you're not, the question has insufficient information.

But if you are, the given figure is drawn wrongly because then $$BD = CD - BC \\= 13 \cos 44^{\circ} - 12 \sin 86^{\circ}$$

which is negative.

If the given sides and angles are correct (and the two angles I mentioned are indeed right angles), then it is clear that point $B$ needs to lie further away from $C$ than $D$ (i.e. instead of lying in between $C$ and $D$ as drawn, $B$ lies on $CD$ produced). In which case, the answer is given by $\displaystyle BD = BC - CD = 12 \sin 86^{\circ} - 13 \cos 44^{\circ} \approx 2.62 \mathrm{cm}$

Answer 3

It's possible if and only if the triangle on the right is a right triangle.

If not, then change point d to a point along the line DB. It is clear that none of the given values change, but DB obviously changes.