Cho hình hộp chữ nhật ABCD.A'B'C'D' có

* Hướng dẫn giải

Đáp án C

Ta có $d\left(D{D}^{\text{'}},A{C}^{\text{'}}\right)=d\left(B{B}^{\text{'}},A{C}^{\text{'}}\right).$

Ta có $A^{\text{'}}{C}^{\text{'}}=\sqrt{{\left(A^{\text{'}}{B}^{\text{'}}\right)}^2+{\left(B^{\text{'}}{C}^{\text{'}}\right)}^2}=2a.$

Kẻ $B^{\text{'}}H\perp A^{\text{'}}{C}^{\text{'}}.$

$B^{\text{'}}H=\frac{A^{\text{'}}{B}^{\text{'}}.B^{\text{'}}{C}^{\text{'}}}{A^{\text{'}}{C}^{\text{'}}}=\frac{a.a\sqrt{3}}{2a}=\frac{a\sqrt{3}}{2}.$

Vì $B{B}^{\text{'}}//\left(AC{C}^{\text{'}}{A}^{\text{'}}\right)$ nên $d\left(B{B}^{\text{'}},A{C}^{\text{'}}\right)=d\left(B{B}^{\text{'}},\left(AC{C}^{\text{'}}{A}^{\text{'}}\right)\right)$

$d\left(B{B}^{\text{'}},\left(AC{C}^{\text{'}}{A}^{\text{'}}\right)\right)=B^{\text{'}}H=\frac{a\sqrt{3}}{2}.$

Nên $d\left(B{B}^{\text{'}},A{C}^{\text{'}}\right)=\frac{a\sqrt{3}}{2}.$