Square root spiral

The governing concept is the Pythagorean theorem applied to the first right triangle of the square-root spiral. Its two perpendicular legs are each 1 unit long. Therefore, if h denotes the hypotenuse, h² = 1² + 1² = 1 + 1 = 2. Taking the positive square root gives h = √2. Consequently, option A correctly explains both the construction and the calculation. Option B has value 4 + 1 = 5, not 2. Option C is an ordinary addition statement and does not establish a hypotenuse. Option D is false because 2 is not a perfect square; indeed, √2 is irrational. The first hypotenuse is therefore √2 precisely because two unit perpendicular sides produce a squared length of 2.

Answer: option C, 1. The defining rule of the square-root spiral is that every newly drawn segment perpendicular to the current radius has unit length. Thus, even though OPₙ = √n, the added segment PₙPₙ₊₁ is always 1. Applying the Pythagorean theorem confirms the purpose of this choice: OPₙ₊₁² = OPₙ² + PₙPₙ₊₁² = n + 1² = n + 1, so the next radius is √(n+1). Option A is the current radius, not the newly added segment. Option B is the square of the current radius, and option D is the square of the next radius. Memory cue: the segment added at every stage is fixed at one unit; only the hypotenuse changes.

The governing concept is that a point labelled √n in the square-root spiral is placed at a distance √n units from the origin. Here n = 9, so the required distance is √9. Because 9 is a perfect square and the principal square root is non-negative, √9 = 3. Therefore the point is 3 units from the origin, making option B correct. Option A is not equal to √9; it is √8, which is a different distance. Option C confuses the radicand 9 with its square root, and option D would correspond to √16 rather than √9. The construction does not change the numerical meaning of the square root; it represents that distance geometrically.

The governing simplification rule is to factor the radicand into a perfect square and a remaining factor. For √50, write 50 = 25 × 2. Then √50 = √(25 × 2) = √25 × √2 = 5√2, because √25 = 5. Therefore option A is correct. Option B would square to 500, not 50, and option C is far too large because it treats 25 as a factor outside the root rather than its square root. Option D corresponds to √20, since (2√5)² = 20. The spiral gives a geometric interpretation of √50 as a distance, but the algebraic simplification remains the same: extract the largest perfect-square factor and leave the square-free factor under the radical.

The governing definition is OPₙ = √n: the subscript identifies the point, while the value gives its distance from the origin. For n = 4, OP₄ = √4 = 2, so the fourth indexed point is 2 units from the origin. Therefore option B is correct. Option A confuses the index 4 with the distance and would incorrectly say the distance is 4. Option C makes an unsupported statement about the number of triangles; the equation says nothing about how many construction steps are shown. Option D is false because 2 is rational, and √4 is a perfect-square root. Keeping the index and the numerical distance conceptually separate is essential when reading labels in the square-root spiral.

The governing rule is extraction of a perfect-square factor from under a radical. Factor 72 as 36 × 2, where 36 is the largest useful perfect-square factor. Then √72 = √(36 × 2) = √36 × √2 = 6√2. Comparing this with a√2 gives a = 6, so option D is correct. Option A would give 3√2, whose square is 18, not 72. Option B gives 4√2, whose square is 32, and option C gives 5√2, whose square is 50. The square-root spiral supplies the geometric meaning of √72 as a distance, but the coefficient is found algebraically by simplifying the radical. Since a is the positive coefficient in the standard form, 6 is the required value.

The governing concept is locating a square root between consecutive whole numbers by comparing the number with nearby perfect squares. Since 8² = 64 and 9² = 81, and 64 < 80 < 81, taking square roots gives 8 < √80 < 9. Therefore, the distance represented by √80 lies between the consecutive integers 8 and 9, so option B is correct. The other options are ruled out because 7² = 49 and 10² = 100; their intervals do not contain 80. The spiral construction may display this length geometrically, but the bounds come from the same perfect-square comparison.