1. **State the problem:** We have a triangle with angles 60°, 75°, and 45°, and the side opposite the 60° angle is 240. We want to find the length of side $x$, which is opposite the 75° angle. 2. **Use the Law of Sines:** The Law of Sines states that in any triangle, $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a$, $b$, and $c$ are sides opposite angles $A$, $B$, and $C$ respectively. 3. **Assign sides and angles:** - Side opposite 60° is 240. - Side $x$ is opposite 75°. 4. **Set up the ratio:** $$\frac{x}{\sin 75^\circ} = \frac{240}{\sin 60^\circ}$$ 5. **Solve for $x$:** $$x = \frac{240 \times \sin 75^\circ}{\sin 60^\circ}$$ 6. **Calculate sine values:** $$\sin 75^\circ = \sin(45^\circ + 30^\circ) = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ = \frac{\sqrt{2}}{2} \times \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \times \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6} + \sqrt{2}}{4} \approx 0.9659$$ $$\sin 60^\circ = \frac{\sqrt{3}}{2} \approx 0.8660$$ 7. **Substitute and compute:** $$x = \frac{240 \times 0.9659}{0.8660} = 240 \times 1.1154 = 267.7$$ **Final answer:** $$x \approx 267.7$$