Let be any positive integer that is not a perfect square

MATHEMATICS · Multi-part · CBSE · Mathematics · The Baudhayana-Pythagoras Theorem · The Baudhayana-Pythagoras Theorem

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. (i) For

n=11

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

\sqrt{11}

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

\sqrt{11}

units. (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.