1. **State the problem:** We need to find the length of the diagonal $AG$ in a cuboid where edges $AB$ and $AD$ meet at a 75° angle, with $AB=23$ cm and $AD=18$ cm. 2. **Understand the geometry:** The diagonal $AG$ connects opposite corners of the cuboid. Since $AB$ and $AD$ form a 75° angle at $A$, and assuming the height $AE$ is perpendicular to the base plane, we can use the law of cosines in the base plane to find the diagonal $AC$, then use the Pythagorean theorem to find $AG$. 3. **Calculate diagonal $AC$ in the base plane:** $$AC^2 = AB^2 + AD^2 - 2 \times AB \times AD \times \cos(75^\circ)$$ Substitute values: $$AC^2 = 23^2 + 18^2 - 2 \times 23 \times 18 \times \cos(75^\circ)$$ Calculate each term: $$23^2 = 529$$ $$18^2 = 324$$ $$2 \times 23 \times 18 = 828$$ $$\cos(75^\circ) \approx 0.2588$$ So: $$AC^2 = 529 + 324 - 828 \times 0.2588 = 853 - 214.3 = 638.7$$ 4. **Calculate $AC$:** $$AC = \sqrt{638.7} \approx 25.27 \text{ cm}$$ 5. **Calculate length $AG$:** Since $AG$ is the space diagonal, and assuming the height $AE$ is the same as $AD$ (18 cm) or if height is not given, we consider the cuboid height equal to $AD$ (18 cm) for this problem. Use Pythagoras in 3D: $$AG = \sqrt{AC^2 + AE^2} = \sqrt{25.27^2 + 18^2}$$ Calculate: $$25.27^2 = 638.7$$ $$18^2 = 324$$ So: $$AG = \sqrt{638.7 + 324} = \sqrt{962.7} \approx 31.03 \text{ cm}$$ **Final answer:** $$\boxed{31.03 \text{ cm}}$$