1. **Problem Statement:** In right triangle $PQR$ with right angle at $Q$, segment $QT$ is perpendicular to hypotenuse $PR$. We need to determine which of the given statements must be true. 2. **Key Concept:** When a perpendicular is drawn from the right angle vertex to the hypotenuse in a right triangle, it creates two smaller right triangles, $PTQ$ and $QTR$, each similar to the original triangle $PQR$ and to each other. 3. **Similarity Relations:** By similarity of triangles, $$\triangle PQR \sim \triangle PTQ \sim \triangle QTR$$ 4. **Corresponding sides ratios:** From similarity, the ratios of corresponding sides are equal: $$\frac{PT}{PQ} = \frac{PQ}{PR} = \frac{TQ}{QR}$$ and $$\frac{TQ}{PQ} = \frac{QR}{PR} = \frac{TR}{TQ}$$ 5. **Geometric Mean Theorem:** The altitude $QT$ to the hypotenuse satisfies: $$QT^2 = PT \times TR$$ 6. **Check given statements:** - $\frac{PT}{TQ} = \frac{TQ}{TR}$ is true by similarity and geometric mean. - $\frac{PT}{TR} = \frac{TR}{TQ}$ is not generally true. - $\frac{PQ}{TQ} = \frac{PR}{PQ}$ is not a standard relation. - $\frac{PQ}{QR} = \frac{QR}{TR}$ is not a standard relation. 7. **Conclusion:** The statement that must be true is: $$\frac{PT}{TQ} = \frac{TQ}{TR}$$ This expresses that $TQ$ is the geometric mean between $PT$ and $TR$. Final answer: $$\boxed{\frac{PT}{TQ} = \frac{TQ}{TR}}$$