Why does (\sqrt{n}) become (\sqrt{n+1}) after adding a (1) unit perpendicular in a square root spiral?

Correct answer: Because ((\sqrt{n})^2+1^2=n+1)

The new side of length 1 is drawn perpendicular to the previous hypotenuse, so the two lengths are the legs of a right triangle. The previous hypotenuse has length \sqrt{n}. Applying the Pythagorean theorem gives the square of the new hypotenuse as (\sqrt{n})^2+1^2=n+1.

Taking the positive square root, because a length is positive, gives the new hypotenuse \sqrt{n+1}. This is why option A is correct. Option B incorrectly treats the new length as \sqrt{n}+1; in general, adding lengths is not the same as adding their squares under a perpendicular construction. The result depends specifically on the right angle and the Pythagorean theorem.