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
Learn more about how Veritas Prep can help you achieve a great GMAT score by checking out their GMAT Prep Options >