To represent √2 on the number line using a square-root spiral, which length is taken in the compass?

Correct answer: The hypotenuse of length √2

The relevant construction principle is that a square-root spiral produces successive hypotenuses of lengths √1, √2, √3, and so forth. To construct √2, begin with two perpendicular unit segments. By the Pythagorean theorem, the resulting hypotenuse has square length 1²+1²=2, so its length is √2. The compass is opened to this constructed hypotenuse and an arc is drawn to transfer that length to the number line. Therefore option A is correct. The √3 hypotenuse belongs to the next stage of the spiral, while a 2-unit perpendicular is not the radius required for this construction and zero length cannot locate √2.