CAPF AC 2023 Q2: General Mental Ability – Quantitative Aptitude
- (a) 25√3 / 6
- (b) 25√3 / 4
- (c) 25√3 / 2
- (d) 25√3
Answer: (b) 25√3 / 4
Explanation
Triangle through the points of contact
Given: three circles of radius 5 cm each touch each other at P, Q and R; find the area of triangle PQR.
How to think: join the three centres first.
When two equal circles touch, their centres are two radii apart, and the point of contact is the midpoint between them.
| Step | Working |
|---|---|
| Distance between any two centres | 5 + 5 = 10 cm |
| Triangle of centres | Equilateral, side 10 cm |
| P, Q, R | Midpoints of its three sides |
| Triangle PQR | Equilateral, side = half of 10 = 5 cm |
| Area | (√3/4) × 5² = 25√3/4 sq. cm |
Check: joining the midpoints of a triangle gives a triangle with one-fourth the area.
The big triangle has area (√3/4) × 100 = 25√3, and a quarter of that is 25√3/4.
Answer: (b) 25√3 / 4.
Why each option is right or wrong
- (a) Incorrect. 25√3/6 does not come from an equilateral triangle of side 5 cm.
- (b) CORRECT. Triangle PQR has side 5 cm, so its area is 25√3/4 sq. cm.
- (c) Incorrect. 25√3/2 is double the correct area.
- (d) Incorrect. 25√3 is the area of the triangle joining the centres, not the contact points.
Key point
Equal circles touching externally: centres are 2r apart; contact points are midpoints. Area of an equilateral triangle = (√3/4) × side². Midpoint triangle = one-fourth of the original.
Remember
Contact points are midpoints: side 5, area 25√3/4.
Common mistakes
- Giving the area of the triangle of centres (25√3) instead of triangle PQR.
- Taking the side of PQR as 10 cm instead of 5 cm.
- Forgetting the factor √3/4 in the area of an equilateral triangle.
Exam tip
Educated guess: the standard formula has /4 in it, and the side here is exactly 5. So (√3/4) × 25 appears directly as option (b).
Sources
- NCERT Class X Mathematics — Circles and Areas Related to Circles
- NCERT Class IX Mathematics — Quadrilaterals (mid-point theorem) and Heron's Formula