Midpoint-parallel segment in a triangle: In ΔABC, D is on AB and E is on AC with DE ∥ BC. Given AD = 8 cm, DB = 4 cm, and AE = 12 cm, find the exact length (in cm) of AC.

Introduction / Context:If a line drawn through a triangle is parallel to the base, it creates a smaller triangle similar to the original. The similarity ratio equals the ratio along a side from the vertex.

Given Data / Assumptions:

• AD = 8 cm, DB = 4 cm ⇒ AB = 12 cm.
• DE ∥ BC ⇒ ΔADE ~ ΔABC.
• AE = 12 cm.

Concept / Approach:Similarity gives AD/AB = AE/AC. Solve for AC using the known lengths.

Step-by-Step Solution:

Verification / Alternative check:Scale factor from small to large triangle is 3/2; lengths are consistent with DE ∥ BC.

Why Other Options Are Wrong:6, 9, 15, 24 are not equal to (3/2)*12.

Common Pitfalls:Using DB/AB instead of AD/AB or mixing which sides correspond under similarity.

Final Answer:18 cm