Geometry Level 2

Quadrilateral $$ABCD$$ has $$AB=CD,$$ and the acute angles $$B$$ and $$C$$ satisfy $$\sin B = \frac{4}{9}$$ and $$\sin C= \frac{5}{6}.$$ Find $\frac{\text{area of }\color{red}{\triangle ABC}}{\text{area of }\color{blue}{\triangle BDC}}.$