Answer
$15,18,21.$
Work Step by Step
1. Let the lower left angle be $A$, use the Law of Cosines, we have $cosA=\frac{(5x)^2+(7x)^2-(6x)^2}{2(5x)(7x)}=\frac{19}{35}$ and $sinA=\frac{12\sqrt 6}{35}$
2. Use the area of the triangle: $\frac{1}{2}(5x)(7x\ sinA)=54\sqrt 6$ or $35x^2(\frac{12\sqrt 6}{35})=2(54\sqrt 6)$, thus $x^2=9$ and $x=3$
3. We have the sides as $15,18,21.$
