In triangle ABC, points D on AB and E on AC satisfy AD = 8 cm, DB = 12 cm, AE = 6 cm and EC = 9 cm. If DE is drawn parallel to side BC, then BC is equal to which multiple of segment DE?

Introduction / Context:
This question tests the concept of similar triangles formed by a line drawn parallel to one side of a triangle. Using the basic proportionality theorem, we can relate side lengths of the smaller and larger triangles to find how many times BC is compared with DE.

Given Data / Assumptions:

• Triangle ABC with D on AB and E on AC
• AD = 8 cm, DB = 12 cm so AB = 20 cm
• AE = 6 cm, EC = 9 cm so AC = 15 cm
• DE ∥ BC
• Need BC expressed as a multiple of DE

Concept / Approach:
If a line segment is drawn parallel to one side of a triangle, joining the other two sides, the two resulting triangles are similar. That means the ratios of corresponding sides are equal. Here, triangle ADE is similar to triangle ABC. The scale factor between them equals AD/AB and also AE/AC. We use this factor to relate BC and DE.

Step-by-Step Solution:
AB = AD + DB = 8 + 12 = 20 cm AC = AE + EC = 6 + 9 = 15 cm Since DE ∥ BC, triangle ADE ~ triangle ABC So AD/AB = AE/AC = DE/BC Compute ratio AD/AB = 8/20 = 2/5 Hence DE/BC = 2/5 So BC/DE = 5/2 Therefore BC = (5/2) DE

Verification / Alternative check:
We can check with a scaled numeric example. Suppose DE were 2 units. Then BC would be (5/2)*2 = 5 units. The ratio of corresponding sides BC:DE is 5:2 which equals AB:AD = 20:8 = 5:2, confirming consistency with similarity properties.

Why Other Options Are Wrong:
2/5 DE and 2/3 DE correspond to DE larger than BC, which conflicts with the full side being longer. 3/2 DE implies BC/DE = 3/2 which does not match AD/AB = 2/5. 4/3 DE also gives a ratio inconsistent with the verified similarity ratio of 5/2.

Common Pitfalls:
Some learners confuse DE/BC with BC/DE and invert ratios incorrectly. Others add lengths incorrectly or forget to check that both AD/AB and AE/AC give the same proportion, which is required if DE is parallel to BC. Mixing up which triangle is larger can also cause reversed ratios.

Final Answer:
Thus, side BC is equal to (5/2) DE.