11.200 Through a point dividing the edge of a regular tetrahedron in the ratio 1 : 4, a plane perpendicular to this edge is drawn. Find the ratio of the volumes of the resulting parts of the tetrahedron.
## Solution.

![](https://cdn.mathpix.com/cropped/2024_05_21_f024bf2ff7725246f3bfg-260.jpg?height=347&width=383&top_left_y=258&top_left_x=100)

Let $A B C D$ be the given regular tetrahedron, $A K: K B=4: 1$ and the section $K L M \perp A B$. Let $E$ be the midpoint of $A B$. Then $K L M \| E C D$, as planes with parallel pairs of intersecting lines $K L$ and $K M, E C$ and $E D$. From the similarity, it follows that

$$
\begin{aligned}
& \frac{V_{B K L M}}{V_{B E C D}}=\left(\frac{B K}{B E}\right)^{3}=\frac{8}{125} \Rightarrow \frac{V_{B K L M}}{V_{A B C D}}=\frac{4}{125} \Rightarrow \\
& \Rightarrow V_{B K L M}: V_{A C D K L M}=\frac{4}{125}:\left(1-\frac{4}{125}\right)=\frac{4}{121}
\end{aligned}
$$

Answer: $\frac{4}{121}$.