1. **Problem statement:** We have a sector OABC of a circle with the center at O. The angle at O (angle AOC) is 60°. The area of the shaded segment ABC is 38 cm². We need to find the perimeter of the shaded segment ABC, which consists of the arc AC plus the chord AC. 2. **Known information:** - Angle AOC = 60° - Area of shaded segment ABC = 38 cm² 3. **Recall formulas:** - Area of sector OAC: $$\text{Area}_{\text{sector}} = \frac{\theta}{360} \times \pi r^2$$ where $\theta = 60°$. - Area of triangle OAC (equilateral triangle since angle 60°): $$\text{Area}_{\triangle} = \frac{1}{2} r^2 \sin 60° = \frac{1}{2} r^2 \times \frac{\sqrt{3}}{2} = \frac{\sqrt{3}}{4} r^2$$ - Area of segment ABC = Area of sector OAC - Area of triangle OAC 4. **Write equation for shaded area:** $$38 = \frac{60}{360} \pi r^2 - \frac{\sqrt{3}}{4} r^2$$ Simplify fraction: $$\frac{60}{360} = \frac{1}{6}$$ Thus, $$38 = \frac{1}{6} \pi r^2 - \frac{\sqrt{3}}{4} r^2 = r^2 \left( \frac{\pi}{6} - \frac{\sqrt{3}}{4} \right)$$ 5. **Solve for $r^2$:} $$r^2 = \frac{38}{\frac{\pi}{6} - \frac{\sqrt{3}}{4}} = \frac{38}{\frac{2\pi - 3\sqrt{3}}{12}} = 38 \times \frac{12}{2\pi - 3\sqrt{3}} = \frac{456}{2\pi - 3\sqrt{3}}$$ 6. **Calculate $r$ approximately:** Use approximate values: $\pi \approx 3.1416$ and $\sqrt{3} \approx 1.732$ Evaluate denominator: $$2\pi - 3\sqrt{3} \approx 2 \times 3.1416 - 3 \times 1.732 = 6.2832 - 5.196 = 1.0872$$ So $$r^2 \approx \frac{456}{1.0872} \approx 419.3$$ $$r \approx \sqrt{419.3} \approx 20.48\ \text{cm}$$ 7. **Find length of arc AC:** Arc length formula: $$l = \frac{\theta}{360} \times 2 \pi r = \frac{60}{360} \times 2 \pi r = \frac{1}{6} \times 2 \pi r = \frac{\pi r}{3}$$ Calculate: $$l = \frac{\pi \times 20.48}{3} \approx \frac{3.1416 \times 20.48}{3} \approx \frac{64.34}{3} \approx 21.45\ \text{cm}$$ 8. **Find length of chord AC:** Chord formula using sine of half angle: $$\text{Chord} = 2r \sin \frac{\theta}{2} = 2 \times 20.48 \times \sin 30° = 40.96 \times 0.5 = 20.48\ \text{cm}$$ 9. **Calculate perimeter of shaded segment ABC:** Perimeter = arc length + chord length $$= 21.45 + 20.48 = 41.93$$ 10. **Final answer:** Rounded to one decimal place: $$\boxed{41.9\ \text{cm}}$$