NCERT Solutions for Class 9th Maths Chapter 3 .5.2 Construction of Length √n — Think and Reflect
Book page 553 Updated on2026-09-08
Q1.
We have seen how to obtain a line whose length is a rational number. How do we obtain lines whose lengths are irrational?
Answer
By using the Baudhāyana–Pythagoras Theorem instead of division. Cutting a unit into q equal parts can only ever produce rational lengths p/q; a right triangle produces lengths that are square roots.
right triangle with legs 1 and 1
hypotenuse² = 1² + 1² = 2
hypotenuse = √2 — an irrational length
The construction of Fig. 3.11 then transfers that length onto the number line:
Step 1. Mark OA = 1 unit on the number line and erect a perpendicular at A.
Step 2. On that perpendicular mark B with AB = 1 unit and join OB. Then OB = √2.
Step 3. With centre O and radius OB draw an arc cutting the number line at P. Then OP = √2, so P is the point √2.
OA = AB = 1, so OB = √2; swinging OB down with a compass marks √2 on the line at P.
Why it happens: a compass moves a length without changing it. The theorem manufactures an irrational length out of two rational ones, and the compass carries it to the axis. This is how the number line gets filled at points no fraction can reach.