1. Problem: To find the length of side BC in two similar triangles ABC and ADE, given |AE|=2, |EC|=3, and |ED|=4. 2. Since triangles ABC and ADE are similar, corresponding sides are proportional. We know AE and EC lie on the same line, so AC = AE + EC = 2 + 3 = 5. 3. The ratio of similarity between triangles ADE and ABC is $$\frac{|AD|}{|AB|} = \frac{|AE|}{|AC|} = \frac{2}{5}$$. 4. We want to find BC. Since BC corresponds to DE in the smaller triangle, the ratio is $$\frac{|DE|}{|BC|} = \frac{2}{5}$$. 5. Given |ED|=4, set up the proportion: $$\frac{4}{|BC|} = \frac{2}{5}$$ 6. Solve for |BC|: $$|BC| = \frac{4 \times 5}{2} = 10$$ 7. Final answer: The length of side BC is 10.