In △ABC, points D on AB and E on AC satisfy AD = 8 cm, BD = 12 cm, AE = 6 cm, EC = 9 cm. Find the ratio BC/DE.

Introduction / Context:If a line through points D on AB and E on AC is parallel to BC, similar triangles imply fixed ratios among segments. We can test proportionality from the given splits to confirm DE ∥ BC, then use the similarity ratio to relate BC and DE.

Given Data / Assumptions:

• AB = AD + DB = 8 + 12 = 20.
• AC = AE + EC = 6 + 9 = 15.
• AD/AB = 8/20 = 2/5; AE/AC = 6/15 = 2/5.

Concept / Approach:Since AD/AB = AE/AC, DE ∥ BC (converse of Basic Proportionality Theorem). Then △ADE ∼ △ABC with scale factor k = AD/AB = 2/5. Thus DE/BC = k, so BC/DE = 1/k = 5/2.

Step-by-Step Solution:

Verification / Alternative check:Ratios from both sides match (2/5), confirming parallelism and the similarity scale.

Why Other Options Are Wrong:Other ratios invert or distort the scale factor; only 5/2 follows from k = 2/5.

Common Pitfalls:Assuming parallelism without checking the side ratios; flipping BC/DE by mistake.

Final Answer:5 2