Cho lăng trụ đứng ABC.A'...

D

- Hướng dẫn giải

Đáp án D

Chọn hệ trục tọa độ như hình vẽ

$\begin{array}{l}A(0;a;a\sqrt{2}),M(\frac{a}{2};0;a\sqrt{2}),B\text{'}(0;0;0),C(a;0;a\sqrt{2})\\ \overrightarrow{AM}(\frac{a}{2};-a;0),\overrightarrow{B\text{'}C}(a;0;a\sqrt{2})\\ \Rightarrow \text{[}\overrightarrow{AM},\overrightarrow{B\text{'}C}\text{]}=(-a^2\sqrt{2};-\frac{a^2}{\sqrt{2}};a^2)\\ \overrightarrow{B\text{'}M}(\frac{a}{2};0;a\sqrt{2})\\ d(AM,B\text{'}C)=\frac{\left|\text{[}\overrightarrow{AM},\overrightarrow{B\text{'}C}\text{]}\overrightarrow{B\text{'}M}\right|}{\left|\text{[}\overrightarrow{AM},\overrightarrow{B\text{'}C}\text{]}\right|}=\frac{a\sqrt{7}}{7}\end{array}$