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In a square root spiral, start from \(\sqrt{18}\). Using the usual construction rule, what is the correct usual construction order (as square roots) to reach \(\sqrt{22}\)?

Answer and explanation

Correct answer: \(\sqrt{18}\rightarrow\sqrt{20}\rightarrow\sqrt{21}\rightarrow\sqrt{22}\)

In the square root spiral, each step increases the hypotenuse from \(\sqrt{n}\) to \(\sqrt{n+1}\) by constructing a right triangle whose legs add up to \(1\) in the sense that the next new hypotenuse corresponds to the next integer under the radical. Therefore, from \(\sqrt{18}\) to \(\sqrt{22}\) you must pass through the consecutive ones: \(\sqrt{19},\sqrt{20},\sqrt{21}\) before reaching \(\sqrt{22}\). Among the given options, the only correct usual consecutive construction order shown ending at \(\sqrt{22}\) is \(\sqrt{18}\rightarrow\sqrt{20}\rightarrow\sqrt{21}\rightarrow\sqrt{22}\).

Related tags

Square Root SpiralClass 9Number SystemsHypotenuseConstruction Order

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