Triangles with a line parallel to base – Using similarity:\nIn ∆ABC, D and E lie on AB and AC respectively such that AD = 8 cm, DB = 12 cm (so AB = 20 cm), AE = 6 cm, EC = 9 cm (so AC = 15 cm). Show that DE ∥ BC and find BC in terms of DE.

Introduction / Context:
When a segment joins points on two sides of a triangle in the same ratio, it is parallel to the third side (Converse of Basic Proportionality Theorem), leading to similar triangles and a fixed scale factor.

Given Data / Assumptions:

• AD/DB = 8/12 = 2/3.
• AE/EC = 6/9 = 2/3.
• Therefore DE ∥ BC.

Concept / Approach:
Similarity of ∆ADE and ∆ABC gives scale factor AD/AB = 8/20 = 2/5. Corresponding sides satisfy DE/BC = 2/5 ⇒ BC = (5/2)DE.

Step-by-Step Solution:

Final Answer:
5/2 DE