1. **State the problem:** We need to find the size of angle $x$ in a right-angled triangle where the opposite side to $x$ is 8.6 cm and the adjacent side to $x$ is 3.8 cm. 2. **Formula used:** To find an angle in a right triangle when opposite and adjacent sides are known, use the tangent function: $$\tan(x) = \frac{\text{opposite}}{\text{adjacent}}$$ 3. **Apply the values:** $$\tan(x) = \frac{8.6}{3.8}$$ 4. **Calculate the ratio:** $$\tan(x) = 2.2631578947$$ 5. **Find the angle $x$ by taking the arctangent (inverse tangent):** $$x = \tan^{-1}(2.2631578947)$$ 6. **Calculate $x$ using a calculator:** $$x \approx 66.3^\circ$$ 7. **Final answer:** The size of angle $x$ is approximately **66.3 degrees** to 1 decimal place.