Square root spiral

\(16=4^2\), \(25=5^2\), and \(36=6^2\) are perfect squares, so their square roots are whole numbers. However, \(38\) is not a perfect square; it lies between \(36\) and \(49\). Therefore, \(\sqrt{38}\) is not a whole number. Exam tip: A square root is a whole number only when the number is a perfect square.

The claim is incorrect. The new right triangle has legs \(\sqrt{7}\) and 1, so by Pythagoras, hypotenuse² = \(7+1=8\); hence it is \(\sqrt{8}\). Exam tip: add the squares of perpendicular sides, not the hypotenuse lengths.

Adding a (1) unit perpendicular to (\sqrt{120}) forms (\sqrt{121}). This is the rule of stepwise construction.

To locate a square root between two numbers, compare the number under the root with their squares. Since \(1^2=1\) and \(2^2=4\), every positive number strictly between 1 and 4 has a square root strictly between 1 and 2. Both 2 and 3 lie in this interval: \(1<2<4\) and \(1<3<4\).

Taking square roots preserves the order for positive numbers, so \(1<\sqrt{2}<2\) and \(1<\sqrt{3}<2\). Neither root equals 2, and neither 2 nor 3 is a perfect square. The statement in option A, written as \(1^2<2,3<2^2\), expresses exactly this comparison. Therefore option A is correct.

In a square root spiral, consecutive hypotenuses have lengths √n and √(n+1). Their squares differ by 1, not their lengths; for example, √5 − 2 is about 0.24. Exam tip: compare squares when checking successive hypotenuses.

The first step of a square root spiral begins with two perpendicular sides, each of length 1. The hypotenuse of this right triangle is the distance from the starting point to the new point. By the Pythagorean theorem, the square of this distance is the sum of the squares of the two perpendicular sides.

Thus, (\sqrt{1})^2+1^2=1+1=2, and the hypotenuse is \sqrt{2}. Therefore option A gives the correct relation. Adding lengths directly, using 2 instead of 1, or subtracting squares would not describe this construction. The right angle is essential because the Pythagorean theorem applies to a right triangle.

(\sqrt{75}) is formed from (\sqrt{74}) and a (1) unit perpendicular. The previous hypotenuse always has one less number.

A square root spiral is formed by adding a unit perpendicular side to each right triangle. By Pythagoras’ theorem, successive hypotenuses are \(\sqrt{2}, \sqrt{3}, \ldots, \sqrt{n}\); hence the point is \(\sqrt{n}\) units from the origin. Exam tip: identify the hypotenuse as the radius.

(\sqrt{64}=8) is a whole number. Adding a (1) unit perpendicular forms the next hypotenuse (\sqrt{65}).

The hypotenuse length of the required square root is taken in the compass. Then an arc is drawn from the origin.

In a square root spiral, successive hypotenuses are \(\sqrt{1}, \sqrt{2}, \sqrt{3}\), and so on. Therefore, the hypotenuse after \(\sqrt{45}\) is \(\sqrt{46}\). Since \(6^2=36\), \(7^2=49\), and \(36<46<49\), \(\sqrt{46}\) lies between \(6\) and \(7\). Although \(\sqrt{47}\) also lies in this interval, it is not the immediate next hypotenuse. Exam tip: compare the number under the root with consecutive perfect squares to find its interval.

A compass is used to take the hypotenuse length and draw an arc on the number line. This is the correct way to transfer the length.

At each step, a unit segment is drawn perpendicular to the previous hypotenuse, making a new right triangle. If the old hypotenuse is \(\sqrt{n}\), Pythagoras gives the next one as \(\sqrt{n+1}\). Exam tip: look for “perpendicular.”

In a square root spiral, \(\sqrt{2}\) is the hypotenuse of a right-angled triangle with two sides of 1 unit. To construct \(\sqrt{3}\), a right-angled triangle is formed using \(\sqrt{2}\) as one side and a perpendicular side of 1 unit; its hypotenuse is \(\sqrt{3}\). Thus, both constructions use Pythagoras’ theorem to obtain the hypotenuse. Option B applies only to \(\sqrt{2}\), not to \(\sqrt{3}\). Exam tip: each new step of the spiral uses the previous hypotenuse and a perpendicular side of 1 unit.

After (\sqrt{15}), (\sqrt{16}) is formed and (\sqrt{16}=4). Therefore the next hypotenuse is the whole number (4).

The governing idea of a square root spiral is the Pythagorean theorem. If the existing hypotenuse is √6 and a perpendicular segment of length 1 is drawn, the new hypotenuse has length √((√6)² + 1²) = √(6 + 1) = √7. Thus option A gives the correct construction. The other choices either add lengths directly or use unsuitable values.

(\sqrt{2}) and (\sqrt{3}) are not whole numbers. Therefore all hypotenuses cannot be whole numbers.

The same hypotenuse length must be taken in the compass. Drawing an arc from the origin gives the correct point.

The governing concept is the Pythagorean theorem in the construction of a square root spiral. If the existing hypotenuse has length √n and a new perpendicular side of length 1 is added, the new hypotenuse h satisfies h²=(√n)²+1²=n+1. Thus h=√((√n)²+1²)=√(n+1), which is the complete and correct form in option A. The expression √n+1 means ordinary addition outside the radical and is generally not equal to √(n+1). Option B has no valid Pythagorean derivation, option C changes addition to subtraction, and option D incorrectly treats n rather than √n as the existing length. Therefore A correctly repairs the shortcut.

First the exact square root length is constructed. Then the same length is marked on the number line using a compass.