i will start you off.

Let be , that is, the width of the rectangle.

Let be , that is, the length of the rectangle.

Let be the perimeter of the darkened region.

Note that the diagonal

we are given ............(1)

by Pythagoras' theorem, .............(2)

you have two simultaneous equations with two unknowns, you can solve for and . the rest should be straight forward

now,

, and you know .

so you can find all the components and hence the perimeter