6. As shown in the figure below, rectangle $A B C D$ is divided into 4 equal-area parts by 5 line segments. Given $X Y=Y B+B C$ $+C Z=Z W=W D+D A+A X, P Q / / A B$. If $B C=19 \mathrm{~cm}, P Q=87 \mathrm{~cm}$, find the length of $A B$ (in cm).
6.18 The trapezoids $X Y Q P$ and $Z W P Q$ have equal areas, $H A Y-\| Z$. Both are equal to $\frac{B C}{2}$. Also, $X Y$ is $\frac{1}{4}$ of the perimeter of rectangle $A B C D$, so
$$
\begin{array}{l}
X Y=\frac{A B+B C}{2} \text { by } \frac{(P Q+X Y)}{2} \times \frac{B C}{2} \\
=\frac{A B \cdot B C}{4} \Rightarrow P Q+X Y=A B \Rightarrow P Q \\
+\frac{A B+B C}{2}=A B \Rightarrow A B=B C+2 P Q \\
=193(\mathrm{~cm}) .
\end{array}
$$