Accidental falls are the number one cause of death and injury in the home. To reduce the chance of falling, make sure all rugs on a hard surface are nonslip.
While you could use double stick carpet tape or a nonslip rug pad to keep them from sliding, a simpler solution is to apply a bead of 100% silicone caulking to the bottom of the rug.
Run the caulking around the edges of the bottom of the rug as well as every 3” or so in the center.
Next, use a putty knife to flatten out the bead of caulking. This will prevent the caulking from being too thick, and allow it to adhere well to the rug.
Let the caulking dry overnight until it has cured before putting the rug in place on the floor. The rubbery texture of the silicone caulking will keep the rug from sliding when you step on it.