1. **Find the area of the shaded sector with radius 6 cm and angle 90°.** The formula for the area of a sector is $$\text{Area} = \pi r^{2} \times \frac{\theta}{360}$$ where $r$ is the radius and $\theta$ the central angle in degrees. Substitute $r = 6$ and $\theta = 90$: $$\text{Area} = \pi \times 6^{2} \times \frac{90}{360} = \pi \times 36 \times \frac{1}{4} = 9\pi \approx 28.27$$ square cm. 2. **Find the area of the shaded segment with radius 10 cm and angle 120°.** The area of a segment is the area of the sector minus the area of the triangular portion formed by the two radii and the chord. - Area of sector: $$\pi r^{2} \times \frac{\theta}{360} = \pi \times 10^{2} \times \frac{120}{360} = \pi \times 100 \times \frac{1}{3} = \frac{100\pi}{3} \approx 104.72$$ square cm. - Area of the triangle formed by the radii (using formula for triangle with two sides and included angle): $$\frac{1}{2} r^{2} \sin \theta = \frac{1}{2} \times 10^{2} \times \sin 120^\circ = 50 \times \sin 120^\circ$$ Since $\sin 120^\circ = \sin (180^\circ - 60^\circ) = \sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.866$, $$\text{Triangle area} = 50 \times 0.866 = 43.3$$ square cm. - Segment area: $$\frac{100\pi}{3} - 43.3 \approx 104.72 - 43.3 = 61.42$$ square cm. 3. **Find the length of an arc with radius 5 cm and central angle 45°.** The length of an arc is given by: $$\text{Arc length} = 2 \pi r \times \frac{\theta}{360}$$ Substitute $r=5$, $\theta=45$: $$\text{Arc length} = 2 \pi \times 5 \times \frac{45}{360} = 10\pi \times \frac{1}{8} = \frac{10\pi}{8} = \frac{5\pi}{4} \approx 3.93$$ cm. **Final answers:** - Area of sector (90° angle, radius 6 cm): approximately 28.27 square cm. - Area of segment (120° angle, radius 10 cm): approximately 61.42 square cm. - Length of arc (45° angle, radius 5 cm): approximately 3.93 cm.