1. **Problem statement:** Find the length of segment $DE$ given the geometric configuration with points $A, B, C, D, E, F, G$ and the following data: $AG=3$, $GB=7$, angles at $C$ are $53^\circ$ and $45^\circ$, $DC=12\sqrt{2}$, $FE=9$, and $CF \perp DE$. The triangle $DEF$ is right-angled. 2. **Understanding the problem:** We have a right triangle $DEF$ with $CF$ perpendicular to $DE$. We know $FE=9$ and $DC=12\sqrt{2}$. The angles at $C$ help us understand the orientation and lengths related to $DC$ and $CF$. 3. **Using trigonometry:** Since $CF$ is perpendicular to $DE$, triangle $DEF$ is right-angled at $F$. We can use the Pythagorean theorem or trigonometric ratios to find $DE$. 4. **Calculate $CF$ using angle $45^\circ$ and length $DC$:** Since $DC=12\sqrt{2}$ and angle $45^\circ$ is involved, we use: $$CF = DC \times \sin 45^\circ = 12\sqrt{2} \times \frac{\sqrt{2}}{2} = 12$$ 5. **Apply Pythagorean theorem in triangle $DEF$:** $$DE = \sqrt{CF^2 + FE^2} = \sqrt{12^2 + 9^2} = \sqrt{144 + 81} = \sqrt{225} = 15$$ 6. **Final answer:** $$DE = 15$$ Thus, the length of segment $DE$ is 15.