Answer:
The minimum possible length of such a line is 8 cm
Explanation:
If we had a rectangle, we can name each side "a" and "b".
The area of the rectangle will be:

Note: This is the constraint of our optimiztion problem.
Applying the Pitagoras theorem, the line, as in the figure attached, will have a length of:

We can replace "a" as a function of "b":

Then,

To calculate the minimum length, we derive and equal to zero:
![dL/db=(d)/(db) [((1024)/(b^2)+b^2)^{(1)/(2)} ]\\\\dL/db=(1)/(2) ((1024)/(b^2)+b^2)^{(-(1)/(2))}\cdot (d)/(db) [(1024)/(b^2)+b^2]\\\\ dL/db=\frac{2b+1024\cdot(-2)\cdot b^(-3)}{2\sqrt{((1024)/(b^2)+b^2)}} \\\\\\ dL/db=\frac{2b-2048\cdot b^(-3)}{2\sqrt{((1024)/(b^2)+b^2)}}](https://img.qammunity.org/2021/formulas/mathematics/college/dz102i50aznb7gm7d564nlvfbmscp03jq5.png)
![dL/db=\frac{2b-2048\cdot b^(-3)}{2\sqrt{((1024)/(b^2)+b^2)}}=0\\\\\\2b-2048b^(-3)=0\\\\2b=(2048)/(b^3)\\\\b^4=(2048)/(2) =1024\\\\b=\sqrt[5]{1024}\approx5.66](https://img.qammunity.org/2021/formulas/mathematics/college/gsy5i45orsiakyv89ezbpzpmtcb6y0ddq5.png)
Now, we know that one side is 5.66 cm.
Then, the other side should be:

The length of the line for this side dimensions will be:
