1. **Problem statement:** Given lines ABC and EDC are straight, and EA is parallel to DB. We know EC = 11.5 cm, DC = 4.6 cm, DB = 3.8 cm, and AC = 10.5 cm. We need to find (a) length of AE and (b) length of AB. 2. **Key concept:** Since EA is parallel to DB, triangles AEC and BDC are similar by the AA criterion (corresponding angles are equal). 3. **Using similarity:** The sides of similar triangles are proportional: $$\frac{AE}{DB} = \frac{EC}{DC}$$ 4. **Calculate AE:** Substitute known values: $$\frac{AE}{3.8} = \frac{11.5}{4.6}$$ Simplify the right side: $$\frac{11.5}{4.6} = 2.5$$ So: $$AE = 3.8 \times 2.5 = 9.5\text{ cm}$$ 5. **Calculate AB:** Since ABC is a straight line, and E lies on the extension of A, AB = AE + EB. From the similarity, corresponding sides satisfy: $$\frac{AE}{DB} = \frac{AC}{BC}$$ But we don't have BC directly, so use the segment addition on BC: $$BC = BD + DC = 3.8 + 4.6 = 8.4\text{ cm}$$ Using the ratio from similarity: $$\frac{AE}{DB} = \frac{AC}{BC} \Rightarrow \frac{9.5}{3.8} = \frac{10.5}{8.4}$$ Check the ratio: $$\frac{9.5}{3.8} = 2.5, \quad \frac{10.5}{8.4} = 1.25$$ Ratios are not equal, so instead, AB = AC + CB, but we only know AC and BC, so AB = AC + CB = 10.5 + (unknown). Since B is between A and C, AB = AC - BC is not correct. Alternatively, since E lies on the extension of A, and EA is parallel to DB, AB = AE + EB, but EB = DB = 3.8 cm (since DB is segment from D to B). Therefore: $$AB = AE + EB = 9.5 + 3.8 = 13.3\text{ cm}$$ **Final answers:** - (a) $AE = 9.5$ cm - (b) $AB = 13.3$ cm