Chad W. answered • 02/03/22
Experienced and Professional Tutor on a Bicycle
The smallest possible ratio is 1 (if both diagonals bisect each other). The largest possible ratio is approached as the short diagonal crosses the very top of the long diagonal, like a capital T. In that case the short sides are 3 cm and the long sides are sqrt(3^2+12^2) = 12.369 (larger than 12), giving a ratio a bit larger than 4.
Intuitively, it seems reasonable that any ratio between 1 and 4.123 is possible. Maybe you've already proven that. If not, you can use the algebraic proof below.
ALGEBRAIC TREATMENT:
The short diagonal is bisected into 3 cm and 3 cm by the long diagonal. The long diagonal is cut into x and (12-x).
Let the short side have length determined by the pythagorean theorm:
sqrt(x^2+3^2)
The long side:
sqrt((12-x)^2+3^2)
Set the ratio.
sqrt((12-x)^2+3^2) / sqrt(x^2+3^2) = 2
Square both sides.
[(12-x)^2+9] / [x^2+9] = 4
Multiply by the denominator.
(12-x)^2+9 = 4x^2+36
Expand the left side.
144-24x+x^2+9 = 4x^2+36
Gather terms.
0 = 3x^2 + 24x - 117
Divide by 3.
0 = x^2 + 8x - 39
Solve with quadratic formula.
x = -4±sqrt(55)
Thus, yes, the sides can have a ratio of 2 to 1. The long diagonal is split into sqrt(55)-4 and 16-sqrt(55), which are approximately 3.416 and 8.584. Using the side-length formulas above, we get side lengths of 4.546 and 9.093, which are in a 1:2 ratio.
