Task A-3.4. (4 points)

Given is a cube $A B C D A_{1} B_{1} C_{1} D_{1}$ with edge length $a$. A plane is defined by its vertices $A$ and $C_{1}$ and the midpoint of the edge $\overline{B B_{1}}$. Calculate the area of the intersection of the cube with this plane.
## Solution.

Let $P$ be the midpoint of edge $\overline{B B_{1}}$, and $Q$ be the midpoint of edge $\overline{D D_{1}}$. Since $A P \| Q C_{1}$ and $P C_{1} \| A Q$, the observed plane intersects edge $\overline{D D_{1}}$ at its midpoint $Q$.

Therefore, the intersection is a rhombus $A P C_{1} Q$, and its area is $P=\frac{1}{2} \cdot\left|A C_{1}\right| \cdot|P Q|$.

It is easy to see that $\left|A C_{1}\right|=a \sqrt{3}$ and $|P Q|=a \sqrt{2}$,

so the area of the rhombus is $P=\frac{1}{2} \cdot\left|A C_{1}\right| \cdot|P Q|=\frac{1}{2} \cdot a \sqrt{3} \cdot a \sqrt{2}=\frac{1}{2} a^{2} \sqrt{6}$.

![](https://cdn.mathpix.com/cropped/2024_05_30_13589bc3b5dd31895527g-15.jpg?height=814&width=779&top_left_y=918&top_left_x=570)

Note: The first point is awarded for explaining the fact that the plane intersects edge $\overline{D D_{1}}$ at its midpoint. A student who has the correct result but has not justified the mentioned statement can receive a maximum of 3 points.

Note: The length of the side of the rhombus $A P C_{1} Q$ is $\frac{a \sqrt{5}}{2}$. A student who determines the length of the side but not the area of the intersection (nor the lengths of the diagonals) will receive a total of 1 or 2 points, depending on whether they have justified the above-mentioned fact.