Question

How do I find the missing side length?

Answer 1

Check the picture below.

$\bf 8\sqrt{6}=a\sqrt{2}\implies \cfrac{8\sqrt{6}}{\sqrt{2}}=a\implies \stackrel{rationalizing~it}{\cfrac{8\sqrt{6}}{\sqrt{2}}\cdot \cfrac{\sqrt{2}}{\sqrt{2}}}=a\implies \cfrac{8\sqrt{12}}{2}=a \\\\\\ 4\sqrt{12}=a\implies 4\sqrt{2^2\cdot 3}=a\implies \boxed{8\sqrt{3}=a}\\\\ -------------------------------$

$\bf 2b=a\implies b=\cfrac{a}{2}\implies b=\cfrac{8\sqrt{3}}{2}\implies \boxed{b=4\sqrt{3}}\\\\ -------------------------------\\\\ x=b\sqrt{3}\implies x=4\sqrt{3}\cdot \sqrt{3}\implies x=4\sqrt{3^2}\implies \boxed{x=12}$