How do you find the length of the sides of a rectangle with only the diagnal length and the area?

Question

With only the area of arectangle and the length of the diagnal, how do you determin the length of the sides?

Answer 1

If the sides are a and b then the area is ab and the length of the diagonal is sqrt(a^2+b^2). So if area=A, then b=A/a and if diagonal length is D then D^2=a^2+b^2=a^2+A^2/a^2. Therefore a^2D^2=a^4+A^2. So a^4-a^2D^2+A^2=0 and a^2=(1/2)(D^2+sqrt(D^4-4A^2)). The two roots of this correspond to a^2 and b^2. To find a and b we just take the positive square root of the two roots (zeroes).

Example: A=60 and D=13. a^2=(1/2)(169+sqrt(28561-14400))=(1/2)(169+sqrt(14161))=(1/2)(169+119)=(1/2)(50) or (1/2)(288)=25 or 144. So a = 5 and b = 12.