Square root spiral

In a square root spiral, the previous hypotenuse becomes one leg of the next right triangle, while the other leg is 1 unit. By Pythagoras, the new hypotenuse squared is the previous square plus 1. Exam tip: every added triangle is right-angled.

Each new right triangle in a square root spiral has one leg of length 1 unit and the previous hypotenuse as the other leg. Hence the squares of hypotenuses increase as 2, 3, 4, …. Exam tip: identify the fixed 1-unit leg.

A square root spiral constructs successive lengths using right triangles. At each step, a new perpendicular side of length 1 is attached to the previously constructed hypotenuse. If the current hypotenuse has length sqrt n, the next one has length sqrt((sqrt n)^2+1^2)=sqrt(n+1) by the Pythagorean theorem. Thus the lengths appear in order as sqrt2,sqrt3,sqrt4,dots.

To construct sqrt15, the immediately preceding step must have produced sqrt14. Adding a unit perpendicular to that length gives sqrt(14+1)=sqrt15. Therefore the length already constructed just before sqrt15 is sqrt14, which is option B. sqrt13 is one step earlier, while sqrt15 is the target itself, not the preceding length.

A square root spiral geometrically constructs lengths like (\sqrt{2},\sqrt{3},\sqrt{4}). It is also useful on the number line.

Pythagoras theorem is used for the hypotenuse in a right triangle. Thus (\sqrt{1^2+1^2}=\sqrt{2}).

The new hypotenuse is formed from previous (\sqrt{n}) and a (1) unit perpendicular. Therefore the previous hypotenuse was (\sqrt{n}).

In a square root spiral, each new 1-unit segment is drawn perpendicular to the previous hypotenuse to form a right triangle. Then the new hypotenuse satisfies h² = n + 1. A parallel segment does not form the required right triangle. Exam tip: check for a 90° angle at every step.

In a square root spiral, the hypotenuse of one triangle becomes one leg of the next right triangle, while the other leg is 1 unit. By Pythagoras, the new hypotenuse becomes \(\sqrt{2}, \sqrt{3}\), and so on. Exam tip: identify the shared side.

At every step of a square root spiral, a perpendicular side of 1 unit is drawn at an end of the previous hypotenuse. By Pythagoras, (new hypotenuse)² = (previous hypotenuse)² + 1. Exam tip: remember that the fixed added side is always 1 unit.

In a square root spiral, the hypotenuse of the previous triangle becomes one leg of the next right triangle, and the other perpendicular leg is 1 unit. By Pythagoras’ theorem, the new hypotenuse becomes \(\sqrt{2}, \sqrt{3}, \sqrt{4}\), and so on. Exam tip: look for the added 1-unit perpendicular side.

A square root spiral starts with a (1) unit line segment. Right triangles are then built from this base.

The two perpendicular sides of the first right triangle are 1 unit each. By Pythagoras’ theorem, hypotenuse² = 1² + 1² = 2, so the hypotenuse is \(\sqrt{2}\) units. Note that 2 is the square of the hypotenuse, not the hypotenuse itself. Exam tip: To find the hypotenuse of a right triangle, add the squares of the perpendicular sides and take the square root.

At each new step a (1) unit perpendicular is drawn on the previous hypotenuse. This makes a new right triangle.

In a square root spiral, the new 1-unit segment is drawn perpendicular to the hypotenuse of the preceding triangle. The resulting hypotenuse represents the next square root. Exam tip: locate the right-angle mark; it is made with the previous hypotenuse.

Adding a (1) unit perpendicular to (\sqrt{6}) gives (\sqrt{7}). The previous hypotenuse has one less number.

The hypotenuses progress as (\sqrt{1},\sqrt{2},\sqrt{3}) and so on. Recognizing the order is useful in easy questions.

In a square root spiral, the previous hypotenuse becomes one side of the next right triangle, and a perpendicular side of 1 unit is added. By Pythagoras, the new hypotenuse gives the next square root. Exam tip: look for the 1-unit perpendicular side.

Each step is based on a right triangle. Therefore Pythagoras theorem is used to find the hypotenuse.

In each new triangle, the previous hypotenuse becomes one side and a 1-unit perpendicular is drawn at its end. By Pythagoras’ theorem, the new hypotenuse represents the next square root. Exam tip: look for the perpendicular unit side.

The previous hypotenuse (\sqrt{9}) with a (1) unit perpendicular gives (\sqrt{10}). Identifying the previous step is important.