AMC10 2008 A

AMC10 2008 A · Q21

AMC10 2008 A · Q21. It mainly tests Area & perimeter, 3D geometry (volume).

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 $ABCD$?

一个边长为1的立方体被一个平面切割，该平面通过两个对角相对的顶点$A$和$C$，以及两条不包含$A$或$C$的对边中点$B$和$D$，如图所示。四边形$ABCD$的面积是多少？

(A) $\frac{\sqrt{6}}{2}$ $\frac{\sqrt{6}}{2}$

(B) $\frac{5}{4}$ $\frac{5}{4}$

(D) $\frac{3}{2}$ $\frac{3}{2}$

(E) $\sqrt{3}$ $\sqrt{3}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

Answer (A): All sides of $ABCD$ are of equal length, so $ABCD$ is a rhombus. Its diagonals have lengths $AC=\sqrt{3}$ and $BD=\sqrt{2}$, so its area is $$\frac{1}{2}\sqrt{3}\cdot\sqrt{2}=\frac{\sqrt{6}}{2}.$$

答案（A）：$ABCD$ 的四条边长度相等，因此 $ABCD$ 是一个菱形。它的两条对角线长度分别为 $AC=\sqrt{3}$ 和 $BD=\sqrt{2}$，所以它的面积为 $$\frac{1}{2}\sqrt{3}\cdot\sqrt{2}=\frac{\sqrt{6}}{2}.$$

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