Because the information" rug covers half of a rectangular floor"; sides of the rug share the same ratio with floor, but we also need to take its total area into account.

In this case, the area of this rug is \cfrac{1}{2} * 9*12 =54. Combining with the ratio, we get

12x^2 = 54 ---->x= \cfrac{3\sqrt2}{2}, so choose the one contains \sqrt2, D is the only option

how did u get 12x square = 54