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ΔABC ~ ΔDEF with AB : DE = 4 : 7. The perimeter of ΔDEF exceeds the perimeter of ΔABC by 18 cm. Find the perimeter of ΔABC.

Class 10Triangles (Similarity)Geometry — Similarity
  • A24 cm
  • B42 cm
  • C18 cm
  • D32 cm

Step-by-Step Solution

Perimeters of similar triangles are in the same ratio as the sides, so take them as and . Then , so and the perimeter of ΔABC = cm. 42 cm is the perimeter of ΔDEF — the other triangle; 18 cm simply repeats the difference; 32 cm comes from using the AREA ratio 16 : 49 where the perimeter ratio 4 : 7 belongs.
Answer: A) 24 cm

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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