Square root spiral

In a square root spiral, each new right triangle has one side of length 1, so its hypotenuse becomes \(\sqrt{2}, \sqrt{3}, \sqrt{4}\), and so on. Hence \(\sqrt{11}\) can be represented. \(\pi\) is not a successive hypotenuse in this construction. Exam tip: look for square roots of non-square natural numbers.

Since 17 is not a perfect square, \(\sqrt{17}\) cannot be expressed as a ratio of two integers. Therefore, this length on the spiral represents an irrational number. Exam tip: first check whether the radicand is a perfect square.

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.

At each step, a new side of length 1 is drawn perpendicular to the previous hypotenuse. By Pythagoras, the new hypotenuse becomes \(\sqrt{n+1}\). The triangles are not equilateral. Exam tip: identify the old hypotenuse and unit side.

Adding a (1) unit perpendicular to (\sqrt{31}) gives (\sqrt{32}). The previous number is (1) less.

For the new right triangle, hypotenuse² = \((\sqrt{7})^2 + 1^2 = 7+1=8\), so the hypotenuse is \(\sqrt{8}\). It is not \(2\sqrt{7}\), as lengths are not simply doubled. Exam tip: apply Pythagoras’ theorem at every step.

In a square root spiral, the hypotenuse of one triangle becomes a side of the next, with a perpendicular side of length 1 added. By Pythagoras, the new hypotenuse squared is the previous hypotenuse squared plus 1. Exam tip: identify the new unit perpendicular side.

Since 10 is not a perfect square, \(\sqrt{10}\) cannot be expressed as a ratio of two integers, so it is irrational. Its spiral length is about 3.16. Exam tip: square roots of perfect squares are rational.

In a square root spiral, each new right triangle is constructed by taking the hypotenuse of the previous triangle as one of its sides. The direction of the new side keeps changing, so the entire construction appears to turn like a spiral. Option B is incorrect because the line segments are not parallel. Exam tip: identify the successive right triangles and their changing direction in a square root spiral.

Adding a (1) unit perpendicular to (\sqrt{49}) gives (\sqrt{50}). Take the previous square root in order.

Taking a (1) unit perpendicular with (\sqrt{11}) gives (\sqrt{12}). This is the step rule of the spiral.

For the new right triangle, hypotenuse² = \((\sqrt{5})^2+1^2=5+1=6\). Hence the new hypotenuse is \(\sqrt{6}\) units. \(\sqrt{10}\) would result if the other side were also \(\sqrt{5}\). Exam tip: square the radical before applying Pythagoras’ theorem.

In a square root spiral, a perpendicular side of length 1 is drawn to the previous hypotenuse. By Pythagoras’ theorem, the new hypotenuse becomes \(\sqrt{2}, \sqrt{3}, \sqrt{4}\), and so on. It is not an equilateral triangle. Exam tip: remember that the added perpendicular side is always 1 unit.

For positive numbers, the square-root function is increasing. Since \(11>10\), we get \(\sqrt{11}>\sqrt{10}\), so Q is farther from the origin. Exam tip: compare the numbers inside the roots first.

(\sqrt{3}) is formed from the sum of squares, not direct addition. The correct calculation is ((\sqrt{2})^2+1^2=3).

A set square helps make a (90^\circ) angle easily. The correct right angle gives the correct hypotenuse.

In a square root spiral, each new hypotenuse is formed using the previous hypotenuse and a perpendicular unit side. Thus, \(\sqrt{5}\) comes from legs \(2\) and \(1\), since \(2^2+1^2=5\). Exam tip: the hypotenuse lies opposite the right angle.

A square root spiral is formed using successive right triangles. The hypotenuse is found by Pythagoras’ theorem, \(c^2=a^2+b^2\), and represents the next square root. Exam tip: identify the hypotenuse first.

In a square root spiral each new right triangle gives the next square root. This is its main idea.

To show the length on the number line, the compass point is placed at the origin and an arc is drawn. This gives the correct square root point.