Square root spiral

A square root spiral begins with a line segment of length 1 unit. A perpendicular segment of 1 unit is then drawn at one end to form the first right triangle, whose hypotenuse is \(\sqrt{2}\). Thus, \(\sqrt{2}\) is obtained after the first construction and is not the starting segment. Exam tip: Remember that the spiral always starts with a 1-unit base.

By Pythagoras theorem the hypotenuse is (\sqrt{1^2+1^2}=\sqrt{2}). The first main hypotenuse is (\sqrt{2}).

In a square root (Theodorus) spiral, each new right triangle is constructed on the previous hypotenuse, with a new perpendicular side of \(1\) unit. By the Pythagorean theorem, the successive hypotenuses become \(\sqrt{2}, \sqrt{3}, \sqrt{4}\), and so on. Thus, \(\sqrt{2}\) and \(\sqrt{3}\) are hypotenuses at early stages, not the new perpendicular side. Exam tip: remember that the side kept fixed in every new triangle is \(1\) unit.

Pythagoras theorem is used to find the hypotenuse. So recognizing the right triangle is important.

Each new triangle is formed by adding a perpendicular side of length 1 to the previous hypotenuse. By Pythagoras, \((\sqrt{n-1})^2+1=n\), so the new hypotenuse is \(\sqrt{n}\). Exam tip: remember the order \(\sqrt2,\sqrt3,\sqrt4\).

With (\sqrt{3}) and a (1) unit perpendicular the new hypotenuse is (\sqrt{4}). Take the next square root in sequence.

The hypotenuses progress as (\sqrt{1},\sqrt{2},\sqrt{3}) and so on. In the spiral the number increases by one.

In a square root spiral, the previous hypotenuse becomes one leg and a 1-unit perpendicular leg is added at its endpoint. By Pythagoras, the next hypotenuse is \(\sqrt{n+1}\). A parallel segment will not form the required right angle. Exam tip: look for the perpendicular mark.

Adding a (1) unit perpendicular to (\sqrt{11}) gives (\sqrt{12}). The previous hypotenuse has one less number.

A square root spiral starts with \(\sqrt{1}\). Each added right triangle increases the square of the hypotenuse by 1, so after 9 new triangles the hypotenuse is \(\sqrt{10}\). Exam tip: count from the initial \(\sqrt{1}\), not from zero.

In a square root spiral, each new right triangle is formed by adding a perpendicular side of 1 unit to the previous hypotenuse. Hence the new hypotenuse becomes \(\sqrt{2}, \sqrt{3}, \sqrt{4}\), and so on. Exam tip: the added side is always 1 unit.

A right angle measures 90°, so 90° is the correct option. In a square root spiral, each new triangle is a right-angled triangle, and the Pythagoras theorem is used to obtain the next length. A 60° angle is associated with an equilateral triangle, not a right angle. Exam tip: whenever you see ‘right angle’, recall 90° immediately.

By Pythagoras the new hypotenuse is (\sqrt{(\sqrt{n})^2+1^2}=\sqrt{n+1}). This is the general rule.

In a square root spiral, the new unit segment is drawn perpendicular to the previous hypotenuse. Thus, \((\sqrt{10})^2+1^2=11\), giving a hypotenuse of \(\sqrt{11}\). A parallel segment does not form the required right triangle. Exam tip: apply Pythagoras at each step.

In a square root spiral, the obtained hypotenuse represents a square-root length. This length is taken in a compass and transferred from 0 onto the number line by drawing an arc. A ruler helps draw or measure lines, but a compass is the appropriate instrument for transferring an exact length. Exam tip: use a compass whenever a construction asks you to transfer a length.

In a square root spiral, a perpendicular side of length 1 unit is drawn on the previous hypotenuse. By Pythagoras’ theorem, the square of the new hypotenuse equals the previous square plus 1. Exam tip: do not confuse the added perpendicular with the hypotenuse.

At each step, a 1-unit side is drawn perpendicular to the previous radius. Thus, hypotenuse² = \(6+1=7\), so the hypotenuse is \(\sqrt{7}\). In exams, always check the right angle.

In a square root spiral the addition is of squares, not direct lengths. The correct reasoning is Pythagoras theorem.

A square-root spiral is formed by attaching successive right triangles, usually with one unit-length perpendicular side, to the preceding hypotenuse. Each new hypotenuse becomes longer and changes direction, so the connected construction winds outward. Consequently, the overall figure resembles a spiral. It is not a straight line, a single square, or merely a circle, making option A correct.

A square root spiral constructs lengths like (\sqrt{2},\sqrt{3},\sqrt{5}). It is useful on the number line.