1. **Problem statement:** The perimeter of triangle $\triangle PQR$ is 94 units, and $QS$ bisects $\angle PQR$. We need to find the lengths of $PS$ and $RS$. 2. **Relevant theorem:** The Angle Bisector Theorem states that the angle bisector divides the opposite side into segments proportional to the adjacent sides. That is, $$\frac{PS}{RS} = \frac{PQ}{QR}$$ 3. **Given values:** - Perimeter $P + Q + R = 94$ - $PQ = 22.4$ - $QR = 29.2$ - $PR = PS + RS$ (unknown segments) 4. **Set variables:** Let $PS = x$ and $RS = y$. Then, $$x + y = PR$$ 5. **Find $PR$:** Since perimeter is sum of all sides, $$PQ + QR + PR = 94$$ $$22.4 + 29.2 + PR = 94$$ $$PR = 94 - 22.4 - 29.2 = 42.4$$ 6. **Apply Angle Bisector Theorem:** $$\frac{x}{y} = \frac{22.4}{29.2}$$ 7. **Express $x$ in terms of $y$:** $$x = y \times \frac{22.4}{29.2}$$ 8. **Use $x + y = 42.4$:** $$y \times \frac{22.4}{29.2} + y = 42.4$$ 9. **Factor out $y$:** $$y \left(\frac{22.4}{29.2} + 1\right) = 42.4$$ 10. **Calculate inside parentheses:** $$\frac{22.4}{29.2} + 1 = \frac{22.4 + 29.2}{29.2} = \frac{51.6}{29.2}$$ 11. **Solve for $y$:** $$y = \frac{42.4 \times 29.2}{51.6}$$ 12. **Calculate $y$:** $$y = \frac{1238.08}{51.6} \approx 23.99$$ 13. **Calculate $x$:** $$x = 23.99 \times \frac{22.4}{29.2} = 23.99 \times 0.7671 \approx 18.41$$ 14. **Final answer:** $$PS \approx 18.41 \text{ units}, \quad RS \approx 23.99 \text{ units}$$