Square root spiral

(\sqrt{2}) is irrational, so its decimal value does not terminate. The spiral represents it exactly geometrically.

The new hypotenuse will be \(\sqrt{3+1}=\sqrt{4}\). Each new right triangle adds \(1^2\).

From (\sqrt{k+1}=\sqrt{21}), we get (k+1=21), so (k=20). In such questions, equate the numbers under the roots.

Then the hypotenuse is (\sqrt{2^2+1^2}=\sqrt{5}), so the standard spiral is not formed. It is necessary to start with (OA=1).

Each new right triangle has the previous hypotenuse as one leg and 1 as the other. By Pythagoras, new square = previous square + 1. Option B is wrong because the length itself does not rise by 1. Exam tip: compare squares of hypotenuses, not their lengths.

Every new (1) unit segment is drawn perpendicular to the previous hypotenuse. This is the correct geometric construction of the spiral.

((\sqrt{7})^2+1^2=8), so the new hypotenuse becomes (\sqrt{8}). The process always starts from the previous square root.

In the square root spiral, (PQ=1) unit is used to form the next triangle. This makes (OQ^2=m+1).

Because the new (1) unit segment is perpendicular to the previous hypotenuse, (OQ^2=OP^2+1). Keep squares and lengths separate while writing the formula.

In a square root spiral, a 1-unit side is drawn perpendicular to the previous hypotenuse. By Pythagoras, if the previous hypotenuse has square n, the new hypotenuse has square n + 1. Exam tip: look for the right-angle mark.

The first triangle gives (\sqrt{2}), so for (\sqrt{12}), the order is (12-1=11). Subtract (1) from the inner number.

The hypotenuse of the (r)th triangle is (\sqrt{r+1}). Therefore, the (10)th triangle has hypotenuse (\sqrt{11}).

In the square root spiral, the labels represent lengths of hypotenuses. The question asks for the difference between the squares of two lengths, not the difference between the lengths themselves. Squaring removes the radical in each case: \\(\sqrt{2})^2=2\\) and \\(\sqrt{3})^2=3\\). Thus the requested difference is obtained by subtracting 2 from 3.

The calculation is \\(3-2=1\\), so option A is correct. Option B, \\(\sqrt{1}\\), has the same numerical value as 1, but it is not the usual listed form of the difference of the squares; option C gives the difference of the two hypotenuse lengths, not their squares. Option D is unrelated. Hence A gives the intended answer.

The start is (OA=1), and then the hypotenuses (\sqrt{2},\sqrt{3},\sqrt{4}) are formed. The number under the root increases continuously.

(\sqrt{27}) is formed by the (26)th triangle, so (26) new perpendicular segments of (1) unit are used. The first (1) unit perpendicular forms (\sqrt{2}).

(\sqrt{16}=4), so it is a rational length. Simplify square roots of perfect squares immediately.

For length (3), the hypotenuse must be (\sqrt{9}). Therefore, it is obtained at the (\sqrt{9}) step.

The new perpendicular side between consecutive hypotenuses is (1) unit. This can also be seen from (PQ^2=31-30=1).

In one step, the number under the root increases only by (1), so (\sqrt{15}) should come after (\sqrt{14}). Skipping a step is incorrect.

In a square root spiral, the new (1) unit segment is drawn perpendicular to the previous hypotenuse. A parallel segment will not allow Pythagoras theorem to apply.