CAPF AC 2025 Q63: General Mental Ability – Quantitative Aptitude
- (a) 1 : π
- (b) √π : 1
- (c) π : 1
- (d) √π : 2
Answer: (c) π : 1
Explanation
Circle around an 18-24-30 triangle
Given: a triangle with sides 18, 24 and 30 cm is inscribed in a circle.
How to think: 18² + 24² = 324 + 576 = 900 = 30², so it is a right-angled triangle with the hypotenuse 30 cm.
The hypotenuse of a right triangle is a diameter of the circle through its vertices.
| Step | Working | Result |
|---|---|---|
| Radius of the circle | 30 ÷ 2 | 15 cm |
| Area of the circle | π × 15² | 225 π cm² |
| Area of the triangle | ½ × 18 × 24 | 216 cm² |
| Ratio | 225 π : 216 | about 1.04 π : 1 |
216 and 225 are close, so the ratio is approximately π : 1.
Answer: (c) π : 1.
Why each option is right or wrong
- (a) Incorrect. 1 : π is the inverse of the ratio.
- (b) Incorrect. √π : 1 uses a square root, which has no place here.
- (c) CORRECT. 225π : 216 is approximately π : 1.
- (d) Incorrect. √π : 2 is far from 225π : 216.
Key point
18-24-30 is a right triangle (3-4-5 scaled by 6). Circumradius = hypotenuse ÷ 2 = 15. Circle area 225π; triangle area 216; ratio ≈ π : 1.
Remember
Right triangle: hypotenuse is the diameter; 225π : 216 ≈ π : 1.
Common mistakes
- Using the area of the triangle as ½ × 30 × 24, with the wrong base and height.
- Taking the radius as 30 instead of 15.
- Expecting an exact ratio when the question says approximately.
Exam tip
Educated guess: 18, 24, 30 is 3, 4, 5 times 6, a right triangle. The circle's area then contains pi, and only (c) and (a) have pi in the right place; the circle is bigger than the triangle, so the ratio is about π : 1.
Sources
- NCERT Class IX Mathematics, Chapter 10 (Circles: angle in a semicircle)
- NCERT Class X Mathematics, Chapter 12 (Areas Related to Circles)