7. [5] Convex quadrilateral $A B C D$ has sides $A B=B C=7, C D=5$, and $A D=3$. Given additionally that $m \angle A B C=60^{\circ}$, find $B D$.
Answer: 8. Triangle $A B C$ is equilateral, so $A C=7$ as well. Now the law of cosines shows that $m \angle C D A=120^{\circ}$; i.e., $A B C D$ is cyclic. Ptolemy's theorem now gives $A C \cdot B D=A B \cdot C D+A D \cdot B C$, or simply $B D=C D+A D=8$.