Question

一个边长为 $1$ 的立方体被一个平面切开，该平面经过两个对角相对的顶点 $A$ 和 $C$ ，以及不含 $A$ 或 $C$ 的两条相对棱的中点 $B$ 和 $D$ ，如图所示。四边形 $ABCD$ 的面积是多少？

A cube with side length $1$ is sliced by a plane that passes through two diagonally opposite vertices $A$ and $C$ and the midpoints $B$ and $D$ of two opposite edges not containing $A$ or $C,$ as shown. What is the area of quadrilateral $ABCD?$

$\dfrac{\sqrt{6}}{2}$

$\dfrac{5}{4}$

$\sqrt{2}$

$\dfrac{3}{2}$

$\sqrt{3}$

答案：A

知识点：正方体菱形对角线

难度评级：1770

解答：

$ABCD$ 的每条边都连接立方体一个顶点和一条棱的中点，所以四边相等， $ABCD$ 是菱形。

它的对角线为立方体空间对角线 $AC = \sqrt{3}$ 和面对角线 $BD = \sqrt{2}$ 。

菱形面积为对角线乘积的一半： $\dfrac{1}{2}\cdot\sqrt{3}\cdot\sqrt{2} = \dfrac{\sqrt{6}}{2}$ 。

所以正确答案是 A。

Each side of $ABCD$ joins a vertex of the cube to the midpoint of an edge, so all four sides are equal and $ABCD$ is a rhombus.

Its diagonals are the space diagonal $AC = \sqrt{3}$ and the face diagonal $BD = \sqrt{2}.$

The area of a rhombus is half the product of its diagonals: $\dfrac{1}{2}\cdot\sqrt{3}\cdot\sqrt{2} = \dfrac{\sqrt{6}}{2}.$

Thus, the correct answer is A.

2008 AMC 10A 第 21 题

答案：A

难度评级：1770

解答：

其他年份的第 21 题