Rectangle ABCD is divided into four parts of equal area by five segments as shown in the figure, where XY=YB+BC+CZ=ZW=WD+DA+AX, and PQ is parallel to AB. Find the length of AB (in cm) if BC=19 cm and PQ=87 cm.[asy]
size(250);
pair A=origin, B=(96,0), C=(96,22), D=(0,22), W=(16,22), X=(20,0), Y=(80,0), Z=(76,22), P=(24,11), Q=(72,11);
draw(P--X--A--D--W--P--Q--Y--B--C--Z--Q^^W--Z^^X--Y);
dot(A^^B^^C^^D^^P^^Q^^W^^X^^Y^^Z);
pair point=(48,11);
label("A", A, SW);
label("B", B, SE);
label("C", C, NE);
label("D", D, NW);
label("P", P, dir(point--P));
label("Q", Q, dir(point--Q));
label("W", W, N);
label("X", X, S);
label("Y", Y, S);
label("Z", Z, N);[/asy] geometryrectangletrapezoid