1. **State the problem**: We have a triangle with angles 100°, 38°, and 42°, and one side length of 13 cm opposite the 42° angle. We need to find the length $x$ cm opposite the 38° angle. 2. **Use the Law of Sines**, which states that in any triangle: $$\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$$ where $a$, $b$, $c$ are side lengths opposite angles $A$, $B$, and $C$ respectively. 3. Assign values: - Side opposite 42° is 13 cm. - Side opposite 38° is $x$ cm. 4. Set up the ratio using Law of Sines: $$\frac{x}{\sin 38^\circ} = \frac{13}{\sin 42^\circ}$$ 5. Solve for $x$: $$x = 13 \times \frac{\sin 38^\circ}{\sin 42^\circ}$$ 6. Calculate the sines (using a calculator): $$\sin 38^\circ \approx 0.6157$$ $$\sin 42^\circ \approx 0.6691$$ 7. Substitute values and compute: $$x \approx 13 \times \frac{0.6157}{0.6691} \approx 13 \times 0.9201 \approx 11.96$$ **Final answer:** $$x \approx 12 \text{ cm}$$