Just before making (\sqrt{30}) in a square root spiral, which hypotenuse will be present?

Correct answer: (\sqrt{29})

A square root spiral is built by repeatedly adding a right triangle with one new perpendicular side of length 1. If the existing hypotenuse has length \(\sqrt{n}\), then the next hypotenuse has length \(\sqrt{n+1}\), because the Pythagorean theorem gives \((\sqrt{n})^2+1^2=n+1\). The construction therefore proceeds in order: \(\sqrt{28}\), \(\sqrt{29}\), \(\sqrt{30}\), and so on.

Just before constructing \(\sqrt{30}\), the existing hypotenuse must be \(\sqrt{29}\). A perpendicular segment of length 1 is then drawn, giving \((\sqrt{29})^2+1^2=29+1=30\), so the new hypotenuse is \(\sqrt{30}\). Hence option B is correct. \(\sqrt{28}\) is one stage earlier, while \(\sqrt{30}\) is the hypotenuse being made, not the one already present.