1. **Problem statement:** We need to find the radius $r$ of the circular arc and then determine the length of the arc from point $E$ to point $F$. 2. **Given data:** - Angle at $E$ is $20^\circ$. - Distance $FG = 30$ mm. - Distance $EG = 200$ mm. 3. **Step to find radius $r$:** The points $E$, $F$, and $G$ lie on the circle with center at the intersection of the radii forming the $20^\circ$ angle. Since $EG$ and $FG$ are radii of the circle, and $EF$ is the chord subtending the $20^\circ$ angle at the center, we can use the Law of Cosines in triangle $EFG$: $$EF^2 = EG^2 + FG^2 - 2 \cdot EG \cdot FG \cdot \cos(20^\circ)$$ But since $EG = FG = r$, this simplifies to: $$EF^2 = r^2 + r^2 - 2r^2 \cos(20^\circ) = 2r^2 (1 - \cos(20^\circ))$$ We know $EF = 30$ mm, so: $$30^2 = 2r^2 (1 - \cos(20^\circ))$$ 4. **Solve for $r$:** $$900 = 2r^2 (1 - \cos(20^\circ))$$ $$r^2 = \frac{900}{2(1 - \cos(20^\circ))}$$ $$r = \sqrt{\frac{900}{2(1 - \cos(20^\circ))}}$$ Calculate $\cos(20^\circ) \approx 0.9397$: $$r = \sqrt{\frac{900}{2(1 - 0.9397)}} = \sqrt{\frac{900}{2(0.0603)}} = \sqrt{\frac{900}{0.1206}} = \sqrt{7464.4} \approx 86.4 \text{ mm}$$ 5. **Find the length of the arc $EF$:** The arc length $s$ is given by: $$s = r \theta$$ where $\theta$ is the central angle in radians. Convert $20^\circ$ to radians: $$\theta = 20^\circ \times \frac{\pi}{180^\circ} = \frac{\pi}{9} \approx 0.3491$$ So, $$s = 86.4 \times 0.3491 \approx 30.15 \text{ mm}$$ **Final answers:** - Radius $r \approx 86.4$ mm - Arc length $EF \approx 30.15$ mm