1. **State the problem:** Given angles $\angle R = 46^\circ$, $\angle S = 85^\circ$, $\angle T = 38^\circ$, and side $t = 17$, find side $s$ using the Law of Sines. 2. **Recall the Law of Sines formula:** $$\frac{s}{\sin 46^\circ} = \frac{17}{\sin 85^\circ}$$ This formula relates the sides and opposite angles of a triangle. 3. **Cross-multiply to solve for $s$:** $$s \sin 85^\circ = 17 \sin 46^\circ$$ 4. **Isolate $s$ by dividing both sides by $\sin 85^\circ$:** $$s = \frac{17 \sin 46^\circ}{\sin 85^\circ}$$ 5. **Calculate the sine values and evaluate:** $$\sin 46^\circ \approx 0.7193, \quad \sin 85^\circ \approx 0.9962$$ 6. **Substitute and compute:** $$s = \frac{17 \times 0.7193}{0.9962} \approx \frac{12.2281}{0.9962} \approx 12.28$$ **Final answer:** $$s \approx 12.28$$