Vector diagram of triangle RQP with labeled segments and vector directions.

- The diagram shows a triangle $PQR$ with points $S$ on side $QP$ and $T$ on side $RP$.
- The segment $ST$ is drawn, and arrows indicate that $ST$ is parallel to $QR$.
- The given lengths are: $QS = 2$, $SP = 6$, $PT = 9$, and $TR = x$.
- Since $ST \parallel QR$, by the Basic Proportionality Theorem (Thales' Theorem), the line $ST$ divides the sides $QP$ and $RP$ proportionally.
- Therefore, the ratio of the segments on one side equals the ratio of the corresponding segments on the other side:
$$
\frac{QS}{SP} = \frac{RT}{TP}
$$
- Substituting the known values:
$$
\frac{2}{6} = \frac{x}{9}
$$
- Simplify the left side:
$$
\frac{1}{3} = \frac{x}{9}
$$
- Solve for $x$ by cross-multiplying:
$$
3x = 9
$$
$$
x = 3
$$