9. 2 (IMO 18) In a planar convex quadrilateral with an area of $32 \mathrm{~cm}^{2}$, the sum of one diagonal and two opposite sides is $16 \mathrm{~cm}$. Determine all possible lengths of the other diagonal.

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9. 2 (IMO 18) In a planar convex quadrilateral with an area of $32 \mathrm{~cm}^{2}$, the sum of one diagonal and two opposite sides is $16 \mathrm{~cm}$. Determine all possible lengths of the other diagonal.
9.2 Let the quadrilateral be $ABCD, AC+AB+CD=16$. Let $BA=b, AC=a, CD=c, \angle BAC=\alpha$. $\angle ACD=\beta$. Then $a+b+c=16, S_{ABCD}=S_{ABC}+S_{ACD}$, which means $64=a(b \sin \alpha+c \sin \beta)$.
$64 \leqslant a(b+c), 64 \leqslant a(16-a)$, we can get $a=8, b+c=8, \alpha=\beta=90^{\circ} . BD=\sqrt{(b+c)^{2}+a^{2}}=8 \sqrt{2} \text{ cm}$