Question 4043
Forgive me:  I haven't learned how to use your equation boxes and draw pictures yet.  Nevertheless, this is an easy one.

Let x = length of the side of the square, where the diagonal of the rectangle is the hypotenuse of the right triangle.

According to the Theorem or Pythagoras, 
a^2 + b^2 = c^2 
x^2 + x^2 = 18^2

Combine like terms on the left side:
2x^2 = 324

Divide both sides by 2
x^2 = 162

Solve for x by taking the square root of each side of the equation.
x = +/- sqrt 162   

Reject the negative answer since a side of a rectangle cannot be negative 
Simplify the square root:
x= sqrt(81) sqrt(2)
x= 9 sqrt(2)