1. **State the problem:** We have two right-angled triangles sharing a side. One triangle has an angle of 35° and side length $x$ opposite this angle. The smaller triangle inside has an angle of 33° and a side length of 5.7 cm opposite this angle. We need to find $x$. 2. **Identify the shared side and use trigonometry:** Let the shared side be $s$. In the smaller triangle, using sine for the 33° angle: $$\sin(33^\circ) = \frac{5.7}{s} \implies s = \frac{5.7}{\sin(33^\circ)}$$ 3. **Calculate $s$:** $$s = \frac{5.7}{\sin(33^\circ)}$$ 4. **Use the larger triangle to find $x$:** The side $x$ is opposite the 35° angle, and $s$ is the hypotenuse of the larger triangle: $$\sin(35^\circ) = \frac{x}{s} \implies x = s \times \sin(35^\circ)$$ 5. **Substitute $s$ from step 3 into step 4:** $$x = \frac{5.7}{\sin(33^\circ)} \times \sin(35^\circ)$$ 6. **Calculate the numerical value:** $$\sin(33^\circ) \approx 0.5446, \quad \sin(35^\circ) \approx 0.5740$$ $$x = \frac{5.7}{0.5446} \times 0.5740 \approx 10.467 \times 0.5740 = 6.0$$ 7. **Final answer:** $$x = 6.0 \text{ cm (to 1 decimal place)}$$