# Problem of the Week Problem C Partitioned Pentagon

Consider pentagon $$PQRST$$. Starting at $$P$$ and moving around the pentagon, the vertices are labelled $$P$$, $$Q$$, $$R$$, $$S$$, and $$T$$, in order.

The pentagon has right angles at $$P$$, $$Q$$, and $$R$$, obtuse angles at $$S$$ and $$T$$, and an area of $$1000\mbox{ cm}^2$$.

Point $$V$$ lies inside the pentagon such that $$\angle PTV$$, $$\angle TVS$$, and $$\angle VSR$$ are right angles.

Point $$U$$ lies on $$TV$$ such that $$\triangle STU$$ has an area of $$210\mbox{ cm}^2$$. Also, it is known that $$PQ=50$$ cm, $$SR=15$$ cm, and $$TU=30$$ cm.

Determine the length of $$PT$$.

Themes: Algebra, Geometry