1. **State the problem:** We need to find the perimeter of a right-angled triangle where one angle is $32^\circ$ and the side adjacent to this angle is $16.6$ m. 2. **Identify the sides:** The side adjacent to the $32^\circ$ angle is $16.6$ m (given). The right angle is at the bottom left, so the side opposite the $32^\circ$ angle and the hypotenuse are unknown. 3. **Use trigonometric ratios:** For a right triangle, - Adjacent side to angle $\theta$ is $a$ - Opposite side is $o$ - Hypotenuse is $h$ We know $a = 16.6$ m and $\theta = 32^\circ$. 4. **Find the hypotenuse using cosine:** $$\cos(32^\circ) = \frac{a}{h} \implies h = \frac{a}{\cos(32^\circ)}$$ Calculate: $$h = \frac{16.6}{\cos(32^\circ)}$$ 5. **Find the opposite side using tangent:** $$\tan(32^\circ) = \frac{o}{a} \implies o = a \times \tan(32^\circ)$$ Calculate: $$o = 16.6 \times \tan(32^\circ)$$ 6. **Calculate values:** $$\cos(32^\circ) \approx 0.8480$$ $$h = \frac{16.6}{0.8480} \approx 19.58$$ $$\tan(32^\circ) \approx 0.6249$$ $$o = 16.6 \times 0.6249 \approx 10.37$$ 7. **Calculate perimeter:** $$P = a + o + h = 16.6 + 10.37 + 19.58 = 46.55$$ 8. **Round to 1 decimal place:** $$P \approx 46.6$$ m **Final answer:** The perimeter of the triangle is $46.6$ m to 1 decimal place.