1. **State the problem:** We have a right triangle with one angle measuring $32^\circ$, one leg measuring 14 units, and the hypotenuse labeled as $x$. We need to find the length of the hypotenuse $x$. 2. **Formula used:** In a right triangle, the cosine of an angle is the ratio of the adjacent side to the hypotenuse: $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}$$ 3. **Identify sides:** The leg measuring 14 units is adjacent to the $32^\circ$ angle, so: $$\cos(32^\circ) = \frac{14}{x}$$ 4. **Solve for $x$:** Multiply both sides by $x$ and then divide both sides by $\cos(32^\circ)$: $$x \cdot \cos(32^\circ) = 14$$ $$\cancel{x} = \frac{14}{\cos(32^\circ)}$$ 5. **Calculate the value:** $$x = \frac{14}{\cos(32^\circ)} \approx \frac{14}{0.8480} \approx 16.51$$ 6. **Conclusion:** The missing side (hypotenuse) $x$ is approximately 16.51 units. None of the provided options (35.4, 19.4, 26.4, 7.4) exactly match this value, but the closest is 19.4 if rounding or measurement error is considered. **Final answer:** $x \approx 16.51$ units.