# Problem of the Week Problem C Locate the Fourth Vertex

Quadrilateral $$BDFH$$ is constructed so that each vertex is on a different side of square $$ACEG$$. Vertex $$B$$ is on side $$AC$$ so that $$AB=4\text{ cm}$$ and $$BC=6\text{ cm}$$. Vertex $$F$$ is on $$EG$$ so that $$EF=3\text{ cm}$$ and $$FG=7\text{ cm}$$. Vertex $$H$$ is on $$GA$$ so that $$GH=4\text{ cm}$$ and $$HA=6\text{ cm}$$. The area of quadrilateral $$BDFH$$ is $$47\text{ cm}^2$$.

The fourth vertex of quadrilateral $$BDFH$$, labelled $$D$$, is located on side $$CE$$ so that the lengths of $$CD$$ and $$DE$$ are both positive integers.

Determine the lengths of $$CD$$ and $$DE$$.