# CAT Practice : Mensuration

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Combo of Mensuration and Geometric Progressions make for an interesting question.

## Squares

Q.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

Choice C. 32 cm2

## Detailed Solution

Diagram -

AP = PB = 2cm

∴ AB= ${\sqrt{(2^2 + 2^2 )}}$ = √8 = 2√2 cm

Similarly, BE = EH = 2√2/2 = √2 cm

∴ EH = ${\sqrt{(√2^2 + √2^2 )}}$

= √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 = ${\frac{a}{1-r}}$

= ${\frac{16}{\frac{1-1}{2}}}$

= 32 cm2

Choice (C) is therefore, the correct answer.

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## More questions from Mensuration

Mensuration is the science of measurement. Cylinders, cones, cuboids, rectangular parallelopipeds etc.