Cho lăng trụ đứng ABC.A'B'C'

B

- Hướng dẫn giải

Đáp án B

Ta có:

$S_{∆MNC}=\frac{S_{∆ABC}}{4}=\frac{a^2}{2}$(đvdt).

$\Rightarrow V_{A\text{'}MNC}=\frac{1}{3}AA\text{'}.S_{∆MNC}=\frac{a^3}{2}$  (đvtt).

Mặt khác: $MN//AB\Rightarrow MN\perp AC$ 

Mà $AA\text{'}\perp mp\left(ABC\right)\Rightarrow MN\perp A{A}^{\text{'}}$

Do đó $S_{∆A\text{'}MN}=\frac{1}{2}A\text{'}M.MN=\frac{1}{2}\sqrt{AA{\text{'}}^2+A{M}^2}$$=\frac{a\sqrt[2]{10}}{2}$   (đvdt).

$\Rightarrow d_{(C;(A\text{'}MN\left)\right)}=\frac{3V_{A\text{'}MNC}}{S_{∆A\text{'}MN}}=\frac{3a}{\sqrt{10}}$  (đvđd).