(a) The sum of length and width is half the perimeter, so the breadth in terms of length (x) in cm will be
.. breadth = 34 -x
(b) Then the expression for the area (in cm^2) is
.. area = length*breadth
.. 253 = (x)(34 -x)
.. x^2 -34x +253 = 0 . . . . . subtract the right side, eliminate parentheses
.. (x -11)(x -23) = 0 . . . . . . . factor
.. x = 11 or 23
The dimensions of the rectangle are 11 cm by 23 cm.