I saw this one posted not too long ago. Lemme see if I can do this correctly.

As you say, the diagonals of the squares show that the rectangle's L:W ratio is 2:1. That means you can solve it so that L = 2W.

Since the perimeter is 18SQRT2, that means

2L + 2W = 18SQRT2
2(2W) + 2W = 18SQRT2
4W + 2W = 18SQRT2
6W = 18SQRT2
W = 3SQRT2

W is equal to the diagonal of the squares. If you know the diagonal of one square, you can find out the length of its side.

If diagonal of a square is 3SQRT2, then the side is 3, since a side of a square is diagonal/SQRT2 -- (This is because a diagonal inside a square makes a 90-45-45 isosceles right triangle - you can use Pythagoreans theorem to prove this)

So if one side of a square is 3, then perimeter of square is 4*3 = 12. Answer choice B.

First, you should realize that the small side of the rectangle is half of the long side.

This is b/c of the 2 inscribed squares.

Therefore, you get: 6x=18sqrt(2)

x=3sqrt(2)

Then, the diagonal of the squares is also 3sqrt(2).

So, a side of the square is 3, b/c the diagonal forms a 45, 45, 90 triangle.

Finally, 3*4=12.

Perimeter equals 12.