Question 26
In the quadrilateral $ABCD$, whose sides are $AB$, $BC$, $CD$ and $DA$, $\angle ABC = \angle BCD = 150^\circ$, $AB = 18\ \text{cm}$ and $BC = 24\ \text{cm}$. Outside the quadrilateral $ABCD$ we draw the equilateral triangles $APB$, $BQC$ and $CRD$. Then we draw the segments $PQ$ and $QR$ and thus a pentagon $APQRD$ is formed. Given that the perimeter of $APQRD$ is $32\ \text{cm}$ greater than the perimeter of $ABCD$, find the length of $CD$.