# CAT Quantitative Aptitude Questions | CAT Geometry - Mensuration Problems

###### CAT Questions | CAT Mensuration Questions | Square and Triangle
CAT Questions

This question is from CAT Mensuration in CAT Geometry.In this question, pictorial reprsentation of a triangle inscribed inside a square is given. And we need to find out the ratio of area. Mensuration Questions in the CAT exam can be solved with practical approach and thought process rather than just knowing theorems from CAT Geometry. Even if one is not completely prepared for CAT Geometry exam, he or she should be able to do well in Mensuration problems tested by the CAT Exam. Knowing Mensuration formulas is important. Make sure you master Mensuration questions for CAT exam.

Question 9: A square PQRS has an equilateral triangle PTO inscribed as shown:

What is the ratio of A∆PQT to A∆TRU?

1. 1 : 3
2. 1 : √3
3. 1 : √2
4. 1 : 2

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### Explanatory Answer

##### Method of solving this CAT Question from Geometry - Mensuration: Find the Ratio of two Triangles inside a square.

Let PQ, a side of equilateral triangle be b
By symmetry QT=ST=z (say)
A ∆ PQT = 1/2 * PQ * QT
= 1/2 * a * z
A ∆ TRU = 1/2 * RT * OR
= 1/2 * (a-z) * (a-z)
∴ $$frac{AΔPQT}{AΔTRU}\\$ = $\frac{1/2∗az}{1/2∗$a−z$^2}$ = $\frac{az}{$a−z$^2}$ --------------$p)
Since PQT and PTO are right angled triangles
PQ2 + QT2 = PT2
RT2 + RU2 = UT2
a2 + z2 = b2 --------- (1)
And, (a-z)2 + (a-z)2 = b2--------- (2)
=> a2 + z2 = 2(a-z)2
=> a2 + z2 = 2a2 + 2z2 – 4az
=> a2 + z2- 4az = 0
=> a2 + z2 – 2az = 2az (Please note how the solution is being managed here. You must always be aware of what you are looking for. Here, as equation -℗ we are looking for (a-z)2 in terms of az)
=> (a-z)2 =2az
Putting in equation (p) = az/2(az) = 1/2

The question is "What is the ratio of A∆PQT to A∆TRU?"

##### Hence, the answer is 1 : 2

Choice D is the correct answer.

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