The law of sines can be used to find the missing side lengths.
... RP = sin(60°)/sin(45°)·10 = 10√(3/2) ≈ 12.247
... PS = sin(75°)/sin(45°)·10 ≈ 13.660
Then the perimeter is
... perimeter = RS + RP + PS
... = 10 + 12.247 + 13.660 ≈ 35.91 . . . .units