Mathematics

Problem

Let

n

be any positive integer that is not a perfect square. A method used in the textbook begins with marking a segment of length

n+1

units on the number line, taking its midpoint, and drawing a semicircle. A perpendicular erected at the point

n

meets the semicircle at P. The perpendicular distance is then transferred to the number line to obtain the point representing

\sqrt{n}

. Answer the following. a. (i) For

n=11

, perform the complete construction on the number line to obtain the point Q representing

\sqrt{11}

. b. (ii) Using geometric reasoning, prove that the length OQ (where O is the origin) equals

\sqrt{11}

units. c. (iii) Generalise the reasoning in (ii) to show that the same construction gives

\sqrt{n}

for every positive integer

n

that is not a perfect square.

Intuitive explanation

This problem uses a classical geometric construction rooted in the geometric mean theorem: when you inscribe a right triangle in a semicircle, the altitude to the hypotenuse creates segments whose geometric mean equals that altitude's length. Here, the segment from 0 to $n+1$ is your hypotenuse, the perpendicular at $n$ splits it into lengths $n$ and $1$ , and the altitude $P$ has length exactly $\sqrt{n \cdot 1} = \sqrt{n}$ . The key insight is recognizing that this perpendicular distance is your answer—you're not calculating anything, just reading off a length that geometry has already constructed for you.

Intuitive explanation

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