4. A cube with side a is intersected by a plane passing through the diagonal of one face and through the midpoints of two adjacent edges of the opposite face. Calculate the area of the intersection.
Solution. The section $A B C D$ (see drawing) is an isosceles trapezoid with bases $\overline{A B}=a \sqrt{2}$, $\overline{C D}=\frac{a \sqrt{2}}{2}$, and leg $\overline{B C}=\frac{a \sqrt{5}}{2}$. For the height $h$ of the trapezoid, we have

$$
h^{2}=\left(\frac{a \sqrt{5}}{2}\right)^{2}-\left(\frac{1}{2}\left(a \sqrt{2}-\frac{a \sqrt{2}}{2}\right)\right)^{2}=\frac{9 a^{2}}{8} .
$$

Thus,

$$
P=\frac{a \sqrt{2}+\frac{a \sqrt{2}}{2}}{2} \frac{a \sqrt{3}}{2 \sqrt{2}}=\frac{3 a^{2} \sqrt{3}}{8}
$$

## Third Year

![](https://cdn.mathpix.com/cropped/2024_06_05_fb2f6c4e736d749cb214g-3.jpg?height=534&width=543&top_left_y=949&top_left_x=940)