직육면체 수선 길이

$\displaystyle \overline{{\mathrm{AE}}} \perp$(평면 $\mathrm{EFGH}$), $\displaystyle \overline{{\mathrm{AI}}} \perp \overline{{\mathrm{FM}}}$이므로 삼수선의 정리에 의하여 $\displaystyle \overline{{\mathrm{EI}}} \perp \overline{{\mathrm{FM}}}$

삼각형 $\mathrm{EFM}$의 넓이에서 $\displaystyle \overline{{\mathrm{EF}}} \times \overline{{\mathrm{FG}}} = \overline{{\mathrm{FM}}} \times \overline{{\mathrm{EI}}}$ 이때 $\displaystyle \mathrm{\overline{FM}} = \sqrt{3 ^{2} + 2 ^{2}} = \sqrt{13}$이므로 $\displaystyle 4 \times 3 = \sqrt{13} \times \mathrm{\overline{EI}}$ $\therefore$ $\displaystyle \mathrm{\overline{EI}} = \frac{12 \sqrt{13}}{13}$ 따라서 직각삼각형 ${\mathrm{EAI}}$에서 $\displaystyle \mathrm{\overline{AI}}$$\displaystyle = \sqrt{\mathrm{\overline{AE}} ^{2} + \mathrm{\overline{EI}} ^{2}}$$\displaystyle = \sqrt{2 ^{2} + \left( \frac{12 \sqrt{13}}{13} \right) ^{2}}$ $\displaystyle = \frac{14 \sqrt{13}}{13}$