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represent root2 on the number line

i) Mark a  number line. Mark a point O on it; then mark OA one unit on this line.

ii) At O erect a perpendicular to the number line; here again mark OB equals one unit.

iii) Join AB; with A as center and AB as radius mark cut off an arc on the number line at C.
So AC equals ?2.

Proof:

Since by construction, OAB is a right triangle, with AB as hypotenuse.
Applying Pythagoras theorem, AB² = OA² + OB²
Each of OA & OB being one unit by construction.
AB² = 1 + 1 = 2
AB = ?2 units.