# CAT Practice : Coordinate Geometry

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Find the ratio of the radius of the circle and the side of the triangle and you are through!!

## Ratio of Areas

Q.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. ${ \Pi : (6 + 4\surd 3)}$
2. ${3\Pi : (6 + 4\surd 3)}$
3. ${2\Pi : (6 + 4\surd 3)}$
4. ${ \Pi : (6 + 2\surd 3)}$

Choice (A).${ \Pi : (6 + 4\surd 3)}$

## Detailed Solution

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

MP = a. AM = a3
AB = 2a + 2a${\surd 3}$
After this we are done.
Area of circle = ${\Pi}$a2
Area of equilateral triangle = ${\frac{√3}{4}}$ * (2a + 2a√3)2
Area of circle = ${\Pi}$a2
Area of equilateral triangle = ${\frac{√3}{4}}$ * (2a + 2a√3)2
= a2(6 + 4√3)
Ratio = ${\Pi}$ : (6 + 4√3)

Correct Answer: (A).${ \Pi : (6 + 4\surd 3)}$

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Geometry is probably the most vital topic as far as CAT preparation is concerned. Geometry sets the stage for Trigonometry, Cogeo and Mensuration as well.