1. **State the problem:** We are given a quadrilateral with points P, Q, R, and S such that angles PQS and RQS are right angles ($90^\circ$). Lengths $PS=5.3$ cm, $PQ=3.8$ cm, and $QR=6.2$ cm are provided. We need to calculate length $RS$ correct to 3 significant figures. 2. **Analyze the figure:** Since both angles at $Q$ are $90^\circ$, lines $PQ$ and $QS$ are perpendicular, similarly $QS$ and $QR$ are perpendicular, meaning $S$ lies such that $PQ \perp QS$ and $QS \perp QR$. This implies $S$ lies at the intersection forming right angles with segments at $Q$. 3. **Use Pythagoras for $PS$:** Observe triangle $PQS$. Since $PQ$ is perpendicular to $QS$ and $PS$ is the hypotenuse of triangle $PQS$, $$PS^2 = PQ^2 + QS^2$$ Substitute values: $$5.3^2 = 3.8^2 + QS^2$$ $$28.09 = 14.44 + QS^2$$ $$QS^2 = 28.09 - 14.44 = 13.65$$ $$QS = \sqrt{13.65} \approx 3.695$$ cm 4. **Use Pythagoras for $RS$:** In triangle $QRS$, since $RQ$ is perpendicular to $QS$, $RS$ is the hypotenuse: $$RS^2 = QR^2 + QS^2$$ $$RS^2 = 6.2^2 + 3.695^2$$ $$RS^2 = 38.44 + 13.65 = 52.09$$ $$RS = \sqrt{52.09} \approx 7.22$$ cm 5. **Final answer:** The length of $RS$ correct to 3 significant figures is $\boxed{7.22}$ cm.