Triangles with a line parallel to base – Using similarity: In ∆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.
Aptitude
Simplification
Difficulty: Easy
Choose an option
Answer
Correct Answer: 5/2 DE
Explanation
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:
AD/DB = AE/EC ⇒ DE ∥ BC.AD/AB = 8/20 = 2/5 ⇒ DE/BC = 2/5 ⇒ BC = (5/2)DE.Final Answer:5/2 DE