[tex]\it ABCD-\ romb,\ \ AC\ \textgreater \ BD,\ \ BD=\dfrac{3}{4}\cdot AC\\\;\\ \\\;\\ \mathcal{A}_{ABCD} =\dfrac{BD\cdot AC}{2} =54 \Rightarrow \dfrac{\dfrac{3}{4}\cdot AC\cdot AC}{2} =54 \Rightarrow \dfrac{3\cdot AC^2}{8} =54 \\\;\\ \\\;\\ \Longrightarrow 3\cdot AC^2 = 8\cdot 54|_{:3} \Longrightarrow AC^2=8\cdot 18 \Longrightarrow AC^2=144 \Longrightarrow \\\;\\ \\\;\\ \Longrightarrow AC = \sqrt{144} \Longrightarrow AC=12 cm[/tex]
[tex]\it BD = \dfrac{3}{4} \cdot AC = \dfrac{3}{4}\cdot12=3\cdot3=9\ cm[/tex]
