1. **State the problem:** We have a right triangle with angles 50°, 90°, and 31°, and one side of length 30 mm opposite the 90° angle. We need to find the missing side $x$ opposite the 31° angle. 2. **Identify the sides and angles:** The side opposite the right angle (90°) is the hypotenuse, so the side of length 30 mm is the hypotenuse. 3. **Use the sine function:** In a right triangle, the sine of an angle is the ratio of the length of the side opposite the angle to the hypotenuse. $$\sin(31^\circ) = \frac{x}{30}$$ 4. **Solve for $x$:** Multiply both sides by 30 to isolate $x$. $$x = 30 \times \sin(31^\circ)$$ 5. **Calculate the sine value:** Using a calculator, $$\sin(31^\circ) \approx 0.5150$$ 6. **Find $x$:** $$x = 30 \times 0.5150 = 15.45$$ 7. **Final answer:** The missing side $x$ is approximately 15.45 mm.