1. **State the problem:** We have a right triangle with a right angle at vertex B, side CB = 30 mm, angle A = 31°, and we need to find the length of side CA (denoted as $x$). 2. **Identify the sides and angles:** - Angle at B = 90° (right angle) - Angle at A = 31° - Side opposite angle B (hypotenuse) = CA = $x$ - Side adjacent to angle A = CB = 30 mm 3. **Use the trigonometric relationship:** Since angle A and the adjacent side CB are known, and we want the hypotenuse CA, use cosine: $$\cos(\text{angle A}) = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{CB}{CA}$$ 4. **Plug in the known values:** $$\cos(31^\circ) = \frac{30}{x}$$ 5. **Solve for $x$:** Multiply both sides by $x$: $$x \cos(31^\circ) = 30$$ Divide both sides by $\cos(31^\circ)$: $$x = \frac{30}{\cos(31^\circ)}$$ Intermediate step showing cancellation: $$x = \frac{30}{\cancel{\cos(31^\circ)}} \times \frac{1}{\cancel{\cos(31^\circ)}}$$ 6. **Calculate the value:** $$x = \frac{30}{\cos(31^\circ)} \approx \frac{30}{0.8572} \approx 35.0$$ 7. **Final answer:** The missing side $x$ is approximately **35.0 mm**.