CAT Quantitative Aptitude Questions | CAT Geometry - Mensuration Problems

This question is from CAT Mensuration in CAT Geometry.In this question, a square is drawn inside a square with some given conditions and it is repeated for infinite times and we need to find the sum of all squares. 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. Combo of Mensuration and Geometric Progressions make for an interesting question.

Question 13: ABCD is a square drawn inside a square PQRS of sides 4 cm by joining midpoints of the sides PQ, QR, RS, SP. Another square is drawn inside ABCD similarly. This process is repeated infinite number of times. Find the sum of all the squares.

1. 16 cm²
2. 28 cm²
3. 32 cm²
4. Infinite

Explanatory Answer

Method of solving this CAT Question from Geometry - Mensuration: Let's go with the given details.

Diagram -

AP = PB = 2cm
∴ AB = √(2² + 2²) = √8 = 2√2 cm

Similarly, BE = EH = \\frac{2√2}{2}\\) = √2 cm
∴ EH = √(√2² + √2²)
= √4 = 2 cm
If we write the infinite series of area of squares,
= 42 + (2√2)² + 2² + ……. infinite
Since it is a decreasing series sum of infinite terms can be approximated.
= 16+8+4+………infinite
∴ a (first term) = 16 , r = \\frac{1}{2}\\) (in the infinite G.P)
Sum of an infinite G.P = \\frac{a}{1 - r}\\)
= \\frac{16}{1-1/2}\\)
= 32 cm²

The question is " Find the sum of all the squares. "

Hence, the answer is 32 cm²

Choice C is the correct answer.