1. **State the problem:** We are given triangle $\triangle PQR$ with perimeter 94 units, and $QS$ bisects $\angle PQR$. We need to find the lengths $PS$ and $RS$. 2. **Recall the Angle Bisector Theorem:** The theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. Mathematically, $$\frac{PS}{SR} = \frac{PQ}{QR}$$ 3. **Identify given values:** - Perimeter: $PQ + QR + PR = 94$ - $PQ = 22.4$ - $QR = 29.2$ - Let $PS = x$ and $RS = y$ so that $PR = x + y$ 4. **Express perimeter in terms of $x$ and $y$:** $$22.4 + 29.2 + (x + y) = 94$$ Simplify: $$51.6 + x + y = 94$$ $$x + y = 94 - 51.6 = 42.4$$ 5. **Use the Angle Bisector Theorem:** $$\frac{x}{y} = \frac{22.4}{29.2}$$ 6. **Solve for $x$ in terms of $y$:** $$x = y \times \frac{22.4}{29.2}$$ 7. **Substitute into the perimeter equation:** $$x + y = 42.4$$ $$y \times \frac{22.4}{29.2} + y = 42.4$$ 8. **Factor out $y$:** $$y \left(\frac{22.4}{29.2} + 1\right) = 42.4$$ 9. **Calculate the sum inside parentheses:** $$\frac{22.4}{29.2} + 1 = \frac{22.4 + 29.2}{29.2} = \frac{51.6}{29.2}$$ 10. **Solve for $y$:** $$y = \frac{42.4 \times 29.2}{51.6}$$ 11. **Calculate $y$:** $$y = \frac{42.4 \times 29.2}{51.6} = \frac{1238.08}{51.6} \approx 23.98$$ 12. **Calculate $x$:** $$x = y \times \frac{22.4}{29.2} = 23.98 \times \frac{22.4}{29.2} = 23.98 \times 0.7671 \approx 18.40$$ **Final answer:** $$PS \approx 18.40 \text{ units}, \quad RS \approx 23.98 \text{ units}$$