# CAT Quantitative Aptitude Questions | Geometry Questions for CAT - Triangles

###### CAT Questions | CAT Geometry Questions | Equilateral Triangle and Square

The question is from CAT Geometry - Triangles. It discusses about an equilateral triangle with a square inscribed inside it. CAT Geometry questions are heavily tested in CAT exam. Make sure you master Geometry problems. This question asks about ratio of area of the square to that of the equilateral triangle.

Question 6: 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. 12 : 12 + 7√3
2. 24 : 24 + 7√3
3. 18 : 12 + 15√3
4. 6 : 6 + 5√3

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### Explanatory Answer

##### Method of solving this CAT Question from Triangles: Mathematicians love symmetry, question setters are nearly besotted with it. So, learn about symmetric figures well.

APQ is an equilateral △. As PQ is parallel to BC.
Let side of the square be ‘a’
AP = a = AQ
△QRC has angles 30 – 60 – 90.

$$frac{QR}{QC}\\$ = $\frac{√3}{2}\\$ QC = QR * $\frac{2}{√3}\\$ AC = AQ + QC = a + $\frac{2a}{√3}\\$ Area of equilateral △ = $\frac{√3}{4}\\$ AC2 = $\frac{√3}{4}\\$$a + $$frac{2a}{√3}\\$)2 = $\frac{√3}{4}\\$$$$frac{√3a + 2a}{√3}\\$)2 = $\frac{√3}{4}\\$ *$$$frac{1}{3}\\$) a2$4 + 3 + 4√3)
= $$frac{7 +4 √3}{12}\\$ * √3a2 = $\frac{12 + 7√3}{12}\\$ * a2 Ratio of area of square to that of equilateral △ is = a2 : $\frac{12 + 7√3}{12}\\$ * a2 = 12 : 12 + 7√3 The question is "What is the ratio of the area of the square to that of the equilateral triangle?" ##### Hence, the answer is 12 : 12 + 7√3. Choice A is the correct answer. ###### Best CAT Online Coaching Try upto 40 hours for free Learn from the best! ###### Best Indore IPM & Rohtak IPM CoachingSignup and sample 9 full classes for free. Register now! ###### Already have an Account? ###### CAT Coaching in ChennaiCAT 2021Enroll at 49,000/- 44,000/- Online Classroom Batches Starting Now! ###### Best CAT Coaching in ChennaiPrices slashed by Rs 4000/- Attend a Demo Class ## CAT Preparation Online | CAT Geometry questions Videos On YouTube #### Other useful sources for Geometry Question | Geometry Triangles Circles Quadrilaterals Sample Questions ##### Where is 2IIM located? 2IIM Online CAT Coaching A Fermat Education Initiative, 58/16, Indira Gandhi Street, Kaveri Rangan Nagar, Saligramam, Chennai 600 093 ##### How to reach 2IIM? Phone:$91) 44 4505 8484
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