Consider hanging them from a wall and taking a photo.

How are you going to get high enough to see the whole thing?
An 8 foot ceiling as found in most homes will require a wide angle
lens. If you stand in the middle you'll have the tripod legs to deal
with. If you shoot from the side your rug will look like a trapezoid
instead of a rectangle.
Is it possible to move the rug? I shoot my wife's quilts by hanging
them on a clothes line. Rugs are probably too heavy for this, but there
are other ways to support them vertically. Then all your problems go
away.

Is this a one time thing, or are you a dealer and need to shoot a dozen or
more a day/week?

for a one time thing, just get up on a ladder  I'd actually do it at an
angle, so one corner is closer, it would be more compositional and look
more, ? active?   If you are a stickler for proportion you could correct any
perspective or keystoning in photoshop.

If this is a periodic thing, get one of those painters ladders that have
bends in the middle so it stretches out like a scaffold.  I don't know if
they make them 12 feet in length, you might want to get two ladders and a
scaffold, this is more than just a board, its has a metal frame to give
serious stiffness.

If you do this daily, run a business as a rug dealer/wholesaler, see if you
can go the crawl space, cut a hold in the ceiling and shoot straight down.
If there is no crawlspace or someone's floor up there, nail a board to the
ceiling, mount a tripod head and hang a camera, there are many cameras with
remote shutter triggering.  mark lines on the floor so you can lay a carpet
out square and you won't have to worry about centering, or focus after the
first time.