PareshGmat wrote:

Square PQRS is inscribed in the square ABCD whose perimeter is four. What is the area of the shaded region:

A: \(\frac{1}{12}\)

B: \(\sqrt{2}/8\)

C: \(\frac{1}{16}\)

D: \(\frac{1}{8}\)

E: \(\sqrt{2}/16\)

There are various ways of approaching it.

Method 1:

Area of ABCD is 1.

Area of PQRS = (1/2)*(diagonal1)*(diagonal2)

Both diagonals of PQRS are 1 each since they are the same lengths as sides of ABCD

Area of PQRS = (1/2)*1*1 = 1/2

Area of leftover region = 1 - 1/2 = 1/2

The leftover region after you cut out PQRS is split into 8 equal areas and the red region is one of those 8.

Hence area of red region is (1/8)*(1/2) = 1/16

Method 2:

The perimeter of ABCD is 4 so each side is 1. So each half side is 1/2.

In triangle APQ, AP and AQ are 1/2 each so \(PQ = 1/\sqrt{2}\) (using Pythegorean theorem)

So half of PQ is \(1/2\sqrt{2}\)

Area of red triangle = \((1/2)*(1/2\sqrt{2})*(1/2\sqrt{2}) = 1/16\)

_________________

Karishma

Veritas Prep GMAT Instructor

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