Given that the areas of the two pentagons are the same, find the sidelength C'D'.?

[varik]

Consider an irregular convex pentagon ABCDE with the following properties:

(1) AB=BC=8,

(2) Angle ABC is a right angle,

(3) CDEA is a square.

Another irregular convex pentagon A'B'C'D'E' has properties:

(1') A'B'=5, B'C'=12,

(2') Angle A'B'C' is a right angle,

(3') C'D'E'A' is a rectangle.

[eoforwic]

Where did you get stuck?
In each case, you divide the pentagon into triangle ABC and rectangle CDEA.
Use the Pythagorean theorem to find AC. This gives you the side of the square and the length of the rectangle.
You have base and height for the two triangles and the side of the square. This lets you find the area of the first pentagon and the triangle in the second. Subtract, and that's the area of the second pentagon's rectangle. Divide by the hypotenuse A'C' (which is 13), and that's the length of C'D'.