1. **State the problem:** We have a triangle with sides labeled $r$, 17, and 5, and an angle of $32^\circ$ opposite side $r$. We need to find the length of side $r$. 2. **Identify the formula:** We can use the Law of Cosines, which states: $$c^2 = a^2 + b^2 - 2ab\cos(C)$$ where $c$ is the side opposite angle $C$. 3. **Assign values:** Let $r = c$, $a = 17$, $b = 5$, and $C = 32^\circ$. 4. **Apply the Law of Cosines:** $$r^2 = 17^2 + 5^2 - 2 \times 17 \times 5 \times \cos(32^\circ)$$ 5. **Calculate each term:** $$17^2 = 289$$ $$5^2 = 25$$ $$2 \times 17 \times 5 = 170$$ 6. **Calculate cosine:** $$\cos(32^\circ) \approx 0.8480$$ 7. **Substitute and simplify:** $$r^2 = 289 + 25 - 170 \times 0.8480$$ $$r^2 = 314 - 144.16$$ $$r^2 = 169.84$$ 8. **Find $r$ by taking the square root:** $$r = \sqrt{169.84} \approx 13.0$$ **Final answer:** $$r \approx 13.0$$