2008 AMC 10A Problem 21

Problem 21 of 25HarderGeometry

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

C,

as shown. What is the area of quadrilateral

ABCD?

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Solution

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.