1. **Stating the problem:** We have a cone with vertex A and base circle with radius FG, and inside it a cylinder with radius CD. Given AG = 12 cm, AC = 2\sqrt{3} cm, and \angle AGE = 60^\circ, we need to find the ratio of the cylinder's radius to the cone's radius, i.e., $\frac{CD}{FG}$. 2. **Understanding the geometry:** AG is the slant height of the cone, AC is a segment inside the cone, and \angle AGE = 60^\circ is the angle at G between points A and E. Points C and D lie on the cylinder's base, and points F and G lie on the cone's base. 3. **Using the given data:** Since AC = $2\sqrt{3}$ cm and AG = 12 cm, and \angle AGE = 60^\circ, we can use trigonometry to find lengths related to the radii. 4. **Finding FG (radius of cone):** Consider triangle AGE. Using the Law of Cosines or trigonometric relations, we find the length GE which corresponds to the radius FG. Since \angle AGE = 60^\circ, and AG = 12 cm, AC = $2\sqrt{3}$ cm, we can find GE using the Law of Cosines in triangle AGE: $$GE^2 = AG^2 + AE^2 - 2 \cdot AG \cdot AE \cdot \cos(60^\circ)$$ But AE is unknown, so instead, we use the right triangle formed by AC and the radius. Since AC is perpendicular to the base, AC is the height of the cylinder inside the cone. 5. **Finding CD (radius of cylinder):** Since AC = $2\sqrt{3}$ cm is the height of the cylinder, and AG = 12 cm is the slant height of the cone, the radius of the cylinder CD can be found by similar triangles or by projecting AC onto the base. 6. **Using similar triangles:** The triangle formed by points A, C, and the base circle is similar to the triangle formed by A, G, and the base circle. The ratio of the radii is proportional to the ratio of the heights: $$\frac{CD}{FG} = \frac{AC}{AG} = \frac{2\sqrt{3}}{12} = \frac{\sqrt{3}}{6} \approx 0.2887$$ 7. **Calculating the ratio:** The ratio $\frac{CD}{FG} \approx 0.2887$. To find the ratio in the form $1 : x$, divide 1 by 0.2887: $$x = \frac{1}{0.2887} \approx 3.46$$ 8. **Checking the options:** None of the options match 1 : 3.46, so let's reconsider the approach. 9. **Alternative approach:** Since \angle AGE = 60^\circ, and AG = 12 cm, the radius FG can be found by: $$FG = AG \cdot \sin(60^\circ) = 12 \cdot \frac{\sqrt{3}}{2} = 6\sqrt{3} \approx 10.39$$ 10. **Radius of cylinder CD:** AC is the height of the cylinder, so the radius CD is: $$CD = AC \cdot \tan(60^\circ) = 2\sqrt{3} \cdot \sqrt{3} = 2 \cdot 3 = 6$$ 11. **Ratio of radii:** $$\frac{CD}{FG} = \frac{6}{6\sqrt{3}} = \frac{1}{\sqrt{3}} \approx 0.577$$ 12. **Expressing ratio as 1 : x:** $$x = \frac{1}{0.577} \approx 1.732$$ 13. **Matching with options:** The closest option is D. 1 : 1.75. **Final answer:** The ratio of the cylinder's radius to the cone's radius is approximately 1 : 1.75.