Square root spiral

Correct answer: B. √223, between 14 and 15

Explanation: Each step of the square root spiral adds 1 to the squared hypotenuse, so a target √224 must be constructed from the previous hypotenuse √223 with a new perpendicular of 1 unit: (√223)² + 1² = 224. To locate the result, compare 224 with consecutive perfect squares. Since 14² = 196 and 15² = 225, we have 196 < 224 < 225. Taking positive square roots gives 14 < √224 < 15. Therefore option B is correct. Option A has the wrong predecessor, option C reverses the correct interval, and option D would apply to √225, not √224.

Correct answer: A. Riya’s marking is incorrect; \(\sqrt{18}\) lies between 4 and 5 and is closer to 4.

Explanation: Since \(4^2=16\) and \(5^2=25\), \(\sqrt{18}\) must lie between 4 and 5. Its value is about 4.24, while \(4.5^2=20.25\). Exam tip: compare the nearest perfect squares first.

Correct answer: B. Take the (\sqrt{2}) hypotenuse length in compass and draw an arc from the origin

Explanation: The same length to be marked is taken in the compass. Drawing an arc from the origin gives the correct position.

Correct answer: B. Adding a perpendicular side of 1 unit to the side of length \(\sqrt{9}\) gives a hypotenuse of length \(\sqrt{10}\).

Explanation: Each new right triangle in the square root spiral has a perpendicular side of 1 unit. \(\sqrt{9^2+1^2}=\sqrt{10}\), so a 10-unit base is not needed. Exam tip: apply Pythagoras' theorem.

Correct answer: A. Because ((\sqrt{4})^2+1^2=5)

Explanation: The direct answer is A. In a square-root spiral, the old hypotenuse becomes one side of a new right triangle, and a perpendicular side of length 1 is added. Here the old length is sqrt{4}. By Pythagoras, the square of the new hypotenuse is (sqrt{4})^2+1^2=4+1=5, so the new hypotenuse is sqrt{5}. Option A is correct because it uses squares of the perpendicular sides. Option B is wrong: lengths are not added directly, since sqrt{4}+1=3, not sqrt{5}. Option C is wrong because 5 is not equal to sqrt{5}. Option D is wrong because the Pythagorean theorem uses addition for a right triangle, not subtraction. Remember: square, add, then take the square root.

Correct answer: B. \(20<\sqrt{440}<21\)

Explanation: \(20^2=400\) and \(21^2=441\). Since \(400<440<441\), taking square roots gives \(20<\sqrt{440}<21\). Although it is close to \(21\), \(\sqrt{440}\) is less than \(21\) because \(440<441\). Exam tip: To locate a square root, compare the number with consecutive perfect squares.

Correct answer: B. (25)-th, (5)

Explanation: The notation in this question directly defines the kth hypotenuse as \\(\\sqrt{k}\\). Therefore, the number under the radical identifies the position of the hypotenuse, while evaluating the radical gives its numerical length. For \\(\\sqrt{25}\\), the index is 25 and the length is 5. Hence option B contains both required parts correctly.

The calculation is \\(\\sqrt{25}=5\\), because 25 is the square of 5. The phrase “25th hypotenuse” refers to the position in the spiral, not to a length of 25. Thus option D confuses the index with the value. The 24th and 26th positions would represent \\(\\sqrt{24}\\) and \\(\\sqrt{26}\\), respectively, so options A and C do not describe the stated segment.

Correct answer: A. The square of the new hypotenuse is \(14+1=15\), so its length is \(\sqrt{15}\).

Explanation: Apply Pythagoras to the new hypotenuse: \(h^2=14+1=15\), so \(h=\sqrt{15}\). Directly adding 1 to \(\sqrt{14}\) is invalid. Exam tip: add squares, not lengths.

Correct answer: B. Each new triangle is a right triangle made from the previous hypotenuse and a (1) unit perpendicular

Explanation: The correct basis of a square root spiral is a right triangle and Pythagoras theorem. Directly adding lengths is wrong.

Correct answer: A. A new unit segment is drawn perpendicular to the previous hypotenuse.

Explanation: At each step, a unit segment is added perpendicular to the previous hypotenuse, creating a right triangle. By Pythagoras,

g(

gsqrt{n})^2+1^2=n+1, so the next hypotenuse is

gsqrt{n+1}. Exam tip: look for “perpendicular” and “unit length.”

Correct answer: B. (24<\sqrt{624}<25)

Explanation: Because (24^2=576) and (25^2=625). The number (624) lies between them, so (\sqrt{624}) lies between (24) and (25).

Correct answer: A. (n+1) is not a perfect square

Explanation: The square root of a positive integer is a whole number only when it is a perfect square. If it is not a perfect square, the square root is irrational.

Correct answer: A. ((\sqrt{20})^2+1^2=21)

Explanation: Pythagoras theorem uses the sum of squares. Therefore (\sqrt{20}) and (1) form hypotenuse (\sqrt{21}).

Correct answer: A. In each new right triangle, one leg is the hypotenuse of the previous triangle and the other leg is 1 unit.

Explanation: Each new right triangle uses the previous hypotenuse as one leg and a unit length as the other, producing \(\sqrt{2}, \sqrt{3}\), and so on. Exam tip: identify the newly added 1-unit leg first.

Correct answer: C. (\sqrt{168}) lies between (13) and (14), (\sqrt{170}) lies between (13) and (14)

Explanation: Because (13^2=169). The number (168) is slightly less, so it is between (12) and (13), while (170) is between (13) and (14).

Correct answer: A. The new perpendicular side should be 1 unit, not √2 units.

Explanation: In a square root spiral, the newly added perpendicular side is always 1 unit. After
√7
, the hypotenuse is
√(7+1)=√8
. Using
√2
would produce
√9
instead. Exam tip: apply the fixed 1-unit side rule at every step.

Correct answer: B. (28<\sqrt{840}<29)

Explanation: Because (28^2=784) and (29^2=841). The number (840) lies between them, so (\sqrt{840}) lies between (28) and (29).

Correct answer: A. The new hypotenuse is (5), and (\sqrt{24}) lies between (4) and (5)

Explanation: After (\sqrt{24}), (\sqrt{25}=5) is formed. Also (4^2<24<5^2), so (\sqrt{24}) lies between (4) and (5).

Correct answer: A. Riya is correct because \(OP=\sqrt{18}\) and \(4^2<18<5^2\).

Explanation: In a square root spiral, the distance OP is \(\sqrt{18}\). Since \(4^2=16\) and \(5^2=25\), \(4<\sqrt{18}<5\). Option B is wrong because \(3^2=9\). Exam tip: compare with nearby perfect squares.

Correct answer: D. \(\frac{3}{2}\)

Explanation: Every radial segment in a square root spiral has length \(\sqrt{n}\), where \(n\) is a positive integer. \(\sqrt{2}\), \(\sqrt{12}\), and \(\sqrt{81}\) fit this form, but \(\frac{3}{2}\) does not. Exam tip: test whether the length can be written as \(\sqrt{n}\).

Correct answer: B. (30<\sqrt{960}<31)

Explanation: Because (30^2=900) and (31^2=961). The number (960) lies between them, so (\sqrt{960}) lies between (30) and (31).

Correct answer: B. It is a successive construction of right triangles where the next hypotenuse is formed by ((\sqrt{n})^2+1^2=n+1)

Explanation: A square root spiral is based on Pythagoras theorem. The previous hypotenuse and (1) unit perpendicular form the next square root.

Correct answer: A. \(\sqrt{50}\) lies between 7 and 8 because \(49<50<64\)

Explanation: Since \(7^2=49\) and \(8^2=64\), \(49<50<64\) gives \(7<\sqrt{50}<8\). Hence A is correct; \(\sqrt{50}\) is not exactly 7. In exams, compare the nearest perfect squares.

Correct answer: A. \(\sqrt{195}\)

Explanation: In a square root spiral, the successive hypotenuses are \(\sqrt{1}, \sqrt{2}, \sqrt{3}, \ldots\). The given new hypotenuse is \(14\), and \(14=\sqrt{196}\). Therefore, the hypotenuse immediately before it is \(\sqrt{195}\). \(\sqrt{196}\) is the current hypotenuse, not the preceding one. Exam tip: Express the whole number as \(\sqrt{n}\), then subtract 1 from the radicand to find the previous hypotenuse.

Correct answer: A. The new hypotenuse will have length \(\sqrt{13}\), because \((\sqrt{12})^2+1^2=13\).

Explanation: In each new right triangle, the previous hypotenuse is one leg and 1 is the other. Thus, new hypotenuse² = 12 + 1 = 13, so its length is \(\sqrt{13}\). Adding \(\sqrt{12}+1\) is incorrect. Exam tip: apply Pythagoras’ theorem.