CAT Quantitative Aptitude Questions | Geometry Questions for CAT - Triangles

CAT Questions

The question is from CAT Geometry - Triangles. In this problem, it discusses about ratio of areas three equal circles that is placed inside an equilateral triangle. CAT Geometry questions are heavily tested in CAT exam. Make sure you master Geometry problems.

Question 24: Three equal circles are placed inside an equilateral triangle such that any circle is tangential to two sides of the equilateral triangle and to two other circles. What is the ratio of the areas of one circle to that of the triangle

1. π : (6+4√3)
2. 3π : (6+4√3)
3. 2π : (6+4√3)
4. π : (6+2√3)

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Explanatory Answer

Method of solving this CAT Question from Triangles: Find the ratio of the radius of the circle and the side of the triangle and you are through!!

Let radii of each circle be ‘a’.
Distance PQ = 2a.
Draw PM perpendicular to AB
∠MAP = 30°

MP = a , AM =√3 a
After this we are done.
Area of circle = πa²
Area of equilateral triangle = \\frac{√3}{4}\\) * (2a + 2a√3)²
= a² (6+4√3)
Ratio = π : (6+4√3)

The question is "What is the ratio of the areas of one circle to that of the triangle?"

Hence, the answer is π : (6+4√3)

Choice A is the correct answer.