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Assertion (A): In ΔABC, D lies on AB and E lies on AC with AD = 2.5 cm, AB = 7.5 cm, AE = 3.2 cm and AC = 9.6 cm; therefore DE ∥ BC. Reason (R): If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side. Choose the correct option.

Class 10Triangles (Similarity)Geometry — Similarity
  • AA is true but R is false
  • BA is false but R is true
  • CBoth A and R are true, and R correctly explains A
  • DBoth A and R are true, but R does not explain A

Step-by-Step Solution

cm and cm, so and . The ratios match, so ∥ and A is true. R is exactly the converse of BPT, the theorem that justifies A, so R is true and explains A.
Answer: C) Both A and R are true, and R correctly explains A

Key idea: Basic Proportionality Theorem (BPT / Thales)

**Theorem**: If a line is drawn parallel to one side of a triangle, intersecting the other two sides at distinct points, then it divides them in the same ratio.

**Setup**: ΔABC. Line DE ∥ BC, with D on AB and E on AC.
**Conclusion**: AD/DB = AE/EC.

**Quick proof sketch**:
Draw DE ∥ BC. Triangles ADE and ABC have ∠A common and ∠ADE = ∠ABC (corresponding angles, DE ∥ BC). So ΔADE ~ ΔABC by AA.
Thus AD/AB = AE/AC.
From AD/AB = AE/AC, you can derive AD/DB = AE/EC by simple algebra.

**Converse**: if AD/DB = AE/EC, then DE ∥ BC.

Useful for proving lines parallel and finding lengths in triangles split by parallels.

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