Try to extend this method for constructing line segments of lengths $\sqrt{3}$ and $\sqrt{5}$ using a ruler and a compass. Generalise this method to construct a line segment of any length of the form $\sqrt{n}$, where $n$ is a positive integer. [Page No. 56]

Represent $\sqrt{3}$ on Number Line:

Let's draw the number line

Represent $\sqrt{5}$ on Number Line: Let's draw the number line

Hence, point $P$ is $\sqrt{5}$.

Generalising: to get $\sqrt{n}$, make a right triangle with legs $\sqrt{n-1}$ and 1, because $\sqrt{(n-1)^2} + 1^2 = n$. Repeating the 'add a unit perpendicular' step from a length-1 start produces $\sqrt{2}, \sqrt{3}, \sqrt{4}, \sqrt{5}, \dots$ in turn each new hypotenuse is $\sqrt{n}$ (one more than the last). A compass then transfers any of these to the number line.