# 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 \(Q\).