CAPF AC 2023 Q3: General Mental Ability – Quantitative Aptitude
- (a) n : m
- (b) n² : m²
- (c) (m + n) : n
- (d) (m + n)² : n²
Answer: (b) n² : m²
Explanation
Areas of similar triangles
Given: triangle ABC has ∠B = 90°; D lies on BC so that triangles ABC and DBA are similar; AB : BC = m : n; find area of ABC : area of ABD.
How to think: for similar triangles, the ratio of areas is the square of the ratio of corresponding sides.
So first match the sides correctly.
The order of the letters tells you: A ↔ D, B ↔ B, C ↔ A.
| Step | Working |
|---|---|
| Corresponding sides | AB ↔ DB and BC ↔ BA |
| Ratio of sides | BC : BA = n : m |
| Ratio of areas | (n : m)² = n² : m² |
Check with numbers: let AB = 3 and BC = 4.
Then DB = AB²/BC = 9/4.
Area of ABC = 6 and area of ABD = (1/2) × 3 × 9/4 = 27/8.
6 : 27/8 = 48 : 27 = 16 : 9, which is n² : m².
Answer: (b) n² : m².
Why each option is right or wrong
- (a) Incorrect. n : m is the ratio of the sides; areas need the square.
- (b) CORRECT. Sides are in the ratio n : m, so areas are in the ratio n² : m².
- (c) Incorrect. (m + n) : n has no basis; nothing here is added.
- (d) Incorrect. (m + n)² : n² uses a wrong pair of sides.
Key point
If two triangles are similar, ratio of areas = square of the ratio of corresponding sides. Read correspondence from the order of the vertices: ABC ~ DBA means A↔D, B↔B, C↔A.
Remember
Similar triangles: areas go as the square of the sides.
Common mistakes
- Stopping at the ratio of sides and forgetting to square it.
- Matching the wrong sides by ignoring the order of the letters.
- Inverting the answer to m² : n², though ABC is the larger triangle.
Exam tip
Educated guess: the word area with similar triangles always means a squared ratio, which leaves (b) and (d). Nothing in the figure adds m and n, so (b).
Sources
- NCERT Class X Mathematics — Triangles (areas of similar triangles)
- R. S. Aggarwal, Quantitative Aptitude — Area and Geometry