1. **State the problem:** We need to find the length $x$ in a triangle where one angle is $55^\circ$, the adjacent side to this angle is 4 cm, and the opposite side to the angle is 7 cm. The triangle has a right angle formed by the height from the top vertex to the base. 2. **Identify the triangle parts and formula:** We can use the Law of Cosines or trigonometric ratios. Here, since we have a right angle and an angle of $55^\circ$, we can use trigonometry. 3. **Use the cosine rule or trigonometric ratios:** The side $x$ is opposite the angle $55^\circ$ and adjacent to the side 4 cm. The side 7 cm is the hypotenuse of the right triangle formed by the height. 4. **Apply the cosine rule:** $$x^2 = 7^2 + 4^2 - 2 \times 7 \times 4 \times \cos(55^\circ)$$ 5. **Calculate each term:** $$x^2 = 49 + 16 - 56 \times \cos(55^\circ)$$ 6. **Calculate $\cos(55^\circ)$:** $$\cos(55^\circ) \approx 0.5736$$ 7. **Substitute and simplify:** $$x^2 = 65 - 56 \times 0.5736 = 65 - 32.1216 = 32.8784$$ 8. **Find $x$ by taking the square root:** $$x = \sqrt{32.8784} \approx 5.7$$ 9. **Final answer:** The length $x$ is approximately **5.7 cm** to 1 decimal place.