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ΔABC ~ ΔDEF with AB = 2.5 cm and DE = 3.5 cm. A student writes ar(ΔABC) : ar(ΔDEF) = 5 : 7. What is the correct ratio, and what went wrong?

Class 10Triangles (Similarity)Geometry — Similarity
  • A25 : 49 — the student gave the ratio of the SIDES instead of its square
  • B5 : 7 — the student is correct
  • C10 : 14 — the sides must first be cleared of decimals
  • D49 : 25 — the larger area is always written first

Step-by-Step Solution

The side ratio is , and the ratio of the areas of similar triangles is the SQUARE of the side ratio: . Clearing decimals changes nothing (10 : 14 is the same ratio as 5 : 7), and the order must follow the triangles named, so 49 : 25 is reversed.
Answer: A) 25 : 49 — the student gave the ratio of the SIDES instead of its square

Key idea: Areas of similar triangles

**Theorem**: The ratio of the areas of two similar triangles is equal to the SQUARE of the ratio of their corresponding sides.

If ΔABC ~ ΔDEF with scale factor , then
Area(ABC)/Area(DEF) = .

**Why**: area scales as length × length. If both length and breadth scale by , area scales by .

**Example**: Two similar triangles have corresponding sides in ratio 3:5. Area ratio = 9:25.

**Reverse**: if areas are in ratio 16:25, sides are in ratio 4:5 (square root).

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