dimensions of the two rectangle

Ramalakshmi wrote:The area of the two rectangle is 84 sq m .The difference between the length and the breadth of the first rectangle is a perfect square. The sum of the length and the breadth of the second rectangle gives the same perfect square. Find the dimensions of the two rectangle.

Say, dimension of the first rectangle is a meters by b meters and that of the first rectangle is c meters by d meters.

Hence, ab = cd = 84
And, (a - b) = (c + d) = n², where n is some integer.

Now note that, 84 = 2*2*3*7

After this we have to do some hit and trial to find the sides.
Note that, (28*3) = 84 and (28 - 3) = 25 = 5²
Also, (21*4) = 84 and (21 + 4) = 25 = 5²

Hence, first rectangle is 28 meters by 3 meters and the second one is 21 meters by 3 meters.

Note: The hit-and-trial is not much time consuming as we have only 6 pairs of factors for 84. Hence we have to check each of those 6 pairs such that they follow the given conditions. There is pure algebraic way also, but that would make the solution more complex than required.