Correct placement of values in a triangle

Question

Task In a right angled triangle, the shortest side is 8.0 centimeter. We know that for the smallest angle $\alpha^{\circ}$, $2.4=\frac{1}{\tan{\alpha^{\circ}}}$. Determine the largest side $c$.

Ok, I go by feeling and decide that the opposing cathetus is eight centimeters long, and that $\alpha^{\circ}=\tan^{-1}\left(\frac{1}{2.4}\right)$, $ \alpha^{\circ}$ is the angle opposite the opposing cathetus. Then I calculate the largest side $c$,

$$\alpha^{\circ}=\tan^{-1}\left(\frac{1}{2.4}\right) =22.62^{\circ}\:$$

$$c=\frac{8}{\sin{22.62^{\circ}}}=21 cm$$

I get the correct answer, my problem is that I get it by going by feeling. How should I interpret texts giving me problems with triangles, e.g. how do you find which side in the triangle is the smallest, and depending on the length of the sides in the triangle, which angle that is the smallest - and so on? (And not only placements of values in a right-angled triangle, but in triangles overall).

Answer 1

In a right triangle, $\triangle\text{ABC}$ we know:

$$ \begin{cases} \left|\text{C}\right|^2=\left|\text{A}\right|^2+\left|\text{B}\right|^2\\ \\ \angle\alpha^\circ+\angle\beta^\circ+90^\circ=180^\circ \end{cases}\tag1 $$

Say, the shortest side is $\left|\text{A}\right|=8$ and $\frac{12}{5}=\frac{1}{\tan\angle\alpha^\circ}$, so we get:

$$\tan\angle\alpha^\circ=\frac{\left|\text{B}\right|}{\left|\text{A}\right|}=\frac{\left|\text{B}\right|}{8}=\frac{5}{12}\space\Longleftrightarrow\space\left|\text{B}\right|=\frac{5\cdot8}{12}=\frac{10}{3}\tag2$$

So, we get:

$$\left|\text{C}\right|^2=8^2+\left(\frac{10}{3}\right)^2=\frac{676}{9}\space\implies\space\left|\text{C}\right|=\sqrt{\frac{676}{9}}=\frac{26}{3}\tag3$$