[tex]BN^{2} = \frac{3a^{2}}{2} \Rightarrow \sqrt{BN^{2}} = \sqrt{\frac{3a^{2}}{2}} \Rightarrow BN = \left | a \right |\frac{\sqrt{3}}{\sqrt{2}} = \frac{\left | a \right |\sqrt{3}\sqrt{2}}{\sqrt{2}\sqrt{2}} = \frac{\left | a \right |\sqrt{6}}{2}[/tex]