In the figure above, the length of each edge of the cube is 2. If P, Q, R, and S are the centres of faces ABFE, BCGF, DCGH, and ADHE, respectively, what is the perimeter of quadrilateral PQRS?
A) 4
B) 4 sqrt2
C) 6
D) 4 sqrt 3
E) 8

Imagine standing over this cube and looking directly down on top of it. You’d see a square with another square inside it, like so:

square in square 1

See the special right triangles all over that? How about now?square in square 2


If each side of the larger square is 2, and each vertex of the smaller square is a midpoint of the sides of the large square, then we know we have a bunch of 45-45-90 triangles with legs of length 1. The sides of the smaller, inner square are the hypotenuses of those triangles, and each have a length of \sqrt{2}. Therefore, the perimeter of that inner square is 4\sqrt{2}.

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