MathDB

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]

Problem Statement