# Straight From P To Q. Well, Not Really

Geometry Level 4

Consider a quadrilateral $$ABCD$$ with $$AB=16$$, $$BC=8$$, $$CD=8$$ and $$\angle ABC = \angle BCD = 90^{\circ}$$. Let $$P$$ be a point on $$AB$$ such that $$AP=2$$ and let $$Q$$ be a point on $$CD$$ such that $$DQ=3$$.

Find the length of the shortest path which:

• Begins at $$P$$, then
• Meets the side $$DA$$ at a point $$W$$, then
• Meets the side $$BC$$ at a point $$X$$, then
• Meets the side $$DA$$ again at a point $$Y$$, then
• Meets the side $$AB$$ at a point $$Z$$, and then
• Ends (finally) at $$Q$$.
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