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.
Aptitude
Area
Difficulty: Easy
Choose an option
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A6 cm
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B18 cm
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C9 cm
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D15 cm
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E24 cm
Answer
Correct Answer: 18 cm
Explanation
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:
AD/AB = 8/12 = 2/3.Thus AE/AC = 2/3 ⇒ AC = (3/2) * AE = (3/2) * 12 = 18 cm.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