1. **State the problem:** We are given a circle with a radius of 4 cm and an arc measuring 70°. We need to find the values of angles A, B, C, and D inside the circle and calculate the area of the 70° sector. 2. **Recall formulas and rules:** - The area of a sector of a circle is given by $$\text{Area} = \frac{\theta}{360} \times \pi r^2$$ where $\theta$ is the central angle in degrees and $r$ is the radius. - The sum of angles around a point inside a circle is 360°. - The inscribed angles subtended by the same arc are equal. 3. **Find angles A, B, C, and D:** - Given the arc measures 70°, the central angle corresponding to this arc is 70°. - Angle A is the central angle, so $A = 70^\circ$. - Angles B and D are inscribed angles subtended by the same arc, so $B = D = \frac{1}{2} A = \frac{1}{2} \times 70 = 35^\circ$. - Angle C is the angle opposite the 70° arc inside the circle, so $C = 180^\circ - 70^\circ = 110^\circ$. 4. **Calculate the area of the 70° sector:** - Use the formula: $$\text{Area} = \frac{70}{360} \times \pi \times 4^2 = \frac{70}{360} \times \pi \times 16$$ - Simplify: $$= \frac{70}{360} \times 16 \pi = \frac{7}{36} \times 16 \pi = \frac{112}{36} \pi = \frac{28}{9} \pi$$ - Approximate: $$\frac{28}{9} \pi \approx 3.111 \times 3.1416 \approx 9.77$$ **Final answers:** - $A = 70^\circ$ - $B = 35^\circ$ - $C = 110^\circ$ - $D = 35^\circ$ - Area of the 70° sector $= \frac{28}{9} \pi \approx 9.77$ square cm