I don't think this is particularly relevant to the solution, but here's a proof that it can't be done (by any means whatsoever) without the line segment of length 1.
If a = 100 cm and b = 100 cm, then the desired line segment has a length 100 * 100 = 10,000.
If a = 1 m and b = 1 m, then the desired line segment has a length 1 * 1 = 1.
But the two conditions are the same, since 1 m = 100 cm. Yet 10,000 cm is not the same as 1 m. What units is the resulting line segment supposed to have? So it is unclear what the problem is supposed to mean, or what units are relevant.
If you add the line segment of length 1, you suddenly know what units you are in, and the problem becomes well-defined.
William, I just wanted to say that this is a great remark!
This remark demonstrates a confusion between linear and square measure. 1 x 1 = 1 metre squared which is 10,000 cm squared. That is a meaning of multiplication here. Try again.