1. **Problem statement:** We have a triangle with points A, B, C on a straight line and a right angle at B. Given angle $\angle A = 20^\circ$, length $AB = 12.6$ cm, and length $DC = 19.3$ cm, we need to find the length $AC$. 2. **Understanding the problem:** Since $ABC$ is a straight line, $AC = AB + BC$. We know $AB$ but need to find $BC$. 3. **Using trigonometry:** In the right triangle $BCD$ (right angle at B), $DC = 19.3$ cm is the hypotenuse. We can find $BC$ using the cosine of angle $20^\circ$ because $\cos 20^\circ = \frac{BC}{DC}$. 4. **Calculate $BC$:** $$BC = DC \times \cos 20^\circ = 19.3 \times \cos 20^\circ$$ 5. **Evaluate $\cos 20^\circ$:** $$\cos 20^\circ \approx 0.9397$$ 6. **Calculate $BC$ numerically:** $$BC = 19.3 \times 0.9397 = 18.13421$$ 7. **Calculate $AC$:** $$AC = AB + BC = 12.6 + 18.13421 = 30.73421$$ 8. **Round to 1 decimal place:** $$AC \approx 30.7 \text{ cm}$$ **Final answer:** The length of $AC$ is approximately $30.7$ cm.