The perimeter of a rectangle in units equals its area in units squared. (4,4), (3,6), and (6,3) are three possible pairs of lengths for this rectangle.

Give another pair of positive integral sides for this rectangle or prove why there isn't another pair.

Let the pairs of lengths be (a, b).

Then, by the problem:

ab = 2(a+b); giving:
Or, ab -2a-2b+4 = 4
Or, xy =4, where (a, b) = (x+2, y+2) .......(#)

Now, the possible integer solutions for xy =4 are:
(x,y) = (1,4); (-1, -4); (2,2); (-2, -2); (4,1); (-4, -1)

Hence, (a, b) = (3,6); (1, -2); (4,4); (0,0); (6,3); (-2, 1)

Since, a, b>0, it follows from (#) that the feasible pairs are given by:

(a,b) = (3,6); (4,4); (6,3)

Posted by K Sengupta on 2007-04-30 13:08:35