# CAT Practice : Coordinate Geometry

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Find the ratio of the side of the square and the side of the triangle and you are through!!

## Ratio of Areas

Q.25: There is an equilateral triangle with a square inscribed inside it. One of the sides of the square lies on a side of the equilateral △. What is the ratio of the area of the square to that of the equilateral triangle?
1. ${ \surd 3 : (5 + 4\surd 3)}$
2. ${2\surd 3 : (7 + 4\surd 3)}$
3. ${4\surd 3 : (7 + 4\surd 3)}$
4. ${ 4\surd 3 : (5 + 2\surd 3)}$

Choice (C).${4\surd 3 : (7 + 4\surd 3)}$

## Detailed Solution

Let side of square be ‘a’.
MN = LP = a
∠ABC = 60°
ML = a. BL = 𝑎${ \frac {a}{√3}}$ (Use some trigonometry for this)

PC = ${ \frac {a}{√3}}$
BC = a + ${ \frac {2a}{√3}}$. We are done.
Area of square = a2
Area of equilateral triangle
= ${ \frac {√3}{4}}$* (a + ${ \frac {2a}{√3}}$)2 = ${ \frac {√3}{4*3}}$ *a2 (√3 + 2)2

= ${ \frac {1}{4*√3}}$ *a2 (7 + 4√3)

Ratio = ${4\surd 3 : (7 + 4\surd 3)}$

Correct Answer: (C). ${4\surd 3 : (7 + 4\surd 3)}$

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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.