9.1. Chords $A A^{\prime}, B B^{\prime}$, and $C C^{\prime}$ of a sphere intersect at a common point $S$. Find the sum $S A^{\prime}+S B^{\prime}+S C^{\prime}$, if $A S=6, B S=3, C S=2$, and the volumes of pyramids $S A B C$ and $S A^{\prime} B^{\prime} C^{\prime}$ are in the ratio $2: 9$. If the answer is not an integer, round it to the hundredths.
Answer: 18. Solution. If $a, b, c$ are the given edges, $x, y, z$ are the corresponding extensions of these edges, and $k$ is the given ratio, then by the theorem of intersecting chords and the lemma on the ratio of volumes of pyramids with equal (vertical) trihedral angles at the vertex, we have: $\left\{\begin{array}{l}a x=b y=c z \\ a b c: x y z=k\end{array} \Rightarrow \sqrt[3]{a b c x y z}=\sqrt[3]{\frac{(a b c)^{2}}{k}} \Rightarrow x=\sqrt[3]{\frac{b^{2} c^{2}}{a k}}\right.$, from which we get $p=x+y+z=x\left(1+\frac{a}{b}+\frac{a}{c}\right)=\sqrt[3]{\frac{b^{2} c^{2}}{a k}}\left(1+\frac{a}{b}+\frac{a}{c}\right)$.

Here $a=6, b=3, c=2, k=\frac{2}{9} \Rightarrow p=\sqrt[3]{\frac{3^{2} \cdot 2^{2} \cdot 9}{6 \cdot 2}}\left(1+\frac{6}{3}+\frac{6}{2}\right)=18$.