3155.61 – Triangle Cleverness

In triangle $\triangle ABC$ (see figure), angle $\angle A$ is $60^\circ$, $\angle B$ is $45^\circ$, and the length of $AC$ is 8. What are the lengths of the other two sides? You can bang out the answers using the Law of Sines, but with those nice angles surely there is another way.

Solution

With the altitude from C drawn, $\triangle CAD$ is a 30-60-90 triangle and $\triangle CDB$ is a 45-45-90 triangle. Hence $AD = 4$ and $CD = 4 \sqrt{3}$. Since $DB$ also is $4 \sqrt{3}$, we find that $CB = 4 \sqrt{6}$ and $AB = 4 + 4 \sqrt{3}$.