Equilateral Area

An equilateral triangle with side length s has area $\frac{s^2 x √3}{4}$ . 9√3 x 4 = 36√3. $\frac{36√3}{√3}$ = 36. √36 = 6

We can use a formula: $A=\dfrac{\sqrt{3}}{4} a^2$ where $a$ is the side length. Substituting

$9\sqrt{3}=\dfrac{\sqrt{3}}{4}x^2$

$x^2=36$

$\boxed{x=6}$

You can split the equalateral triangle into two 30, 60, 90 triangles. From there, we can set the hypotenuse equal to $x$ , the side adjacent to 60 degrees to be $\frac{x}{2}$ and the opposite lenth equal to $\frac{\sqrt{3}x}{2}$ . We can find the area of both triangles: $2 \cdot (\frac{\sqrt{3}x}{2} \cdot \frac{x}{2}/2=\frac{x^{2}\sqrt{3}}{4})$ . We end up with $x^{2}\sqrt{3}=36 \sqrt{3}$ , therefore, $\boxed{x=6}$ .