I hate rounding.
Let's call the diagonal x. It's the hypotenuse of the right triangle whose legs are the rectangle sides.
According to the problem we have a length x-4 and a width x-5 and an area
82 = (x-4)(x-5)
82 = x^2 - 9x + 20
0 = x^2 - 9x - 62
That one doesn't seem to factor so we go to the quadratic formula
Only the positive value makes any sense for this problem, so we conclude
That's the exact answer. Did I mention I hate rounding? That's about
x = 13.6 meters
Answer: 13.6
----------
It's not clear to me this problem is consistent. By the Pythagorean Theorem the diagonal satisfies
which works out to
That's not consistent with the first answer; this problem really has no solution. Tell your teacher to get better material.