1. The problem involves understanding and solving questions related to chords, lengths, arcs, and segments in circles, which are common topics in geometry. 2. Key formulas and concepts include: - Length of a chord: $c = 2r \sin\left(\frac{\theta}{2}\right)$ where $r$ is the radius and $\theta$ is the central angle in radians. - Length of an arc: $L = r\theta$ where $\theta$ is in radians. - Area of a segment: $A = \frac{r^2}{2}(\theta - \sin\theta)$. 3. Important rules: - Angles must be in radians for these formulas. - The central angle $\theta$ corresponds to the arc or chord. 4. Example: Given a circle with radius $r=7$ cm and a central angle $\theta=60^\circ = \frac{\pi}{3}$ radians, find the chord length. 5. Calculate chord length: $$c = 2 \times 7 \times \sin\left(\frac{\pi}{6}\right) = 14 \times \frac{1}{2} = 7 \text{ cm}$$ 6. Calculate arc length: $$L = 7 \times \frac{\pi}{3} = \frac{7\pi}{3} \approx 7.33 \text{ cm}$$ 7. Calculate segment area: $$A = \frac{7^2}{2} \left( \frac{\pi}{3} - \sin\frac{\pi}{3} \right) = \frac{49}{2} \left( \frac{\pi}{3} - \frac{\sqrt{3}}{2} \right) \approx 24.5 \times (1.047 - 0.866) = 24.5 \times 0.181 = 4.44 \text{ cm}^2$$ 8. These calculations help solve many problems involving chords, arcs, and segments in circles.