(i) √5

We can write 5 as the sum of the squares of two natural numbers: 5 = 1 + 4

⇒ 5 = 1² + 2²

On the number line, take OA = 2 units.

Draw BA = 1 units, perpendicular to OA. Join OB.

By Pythagoras theorem, OB = √5 Using a compass with centre O and radius OB, draw an arc which intersects the number line at the point C. Then, C corresponds to √5.

(ii) √10

We can write 10 as the sum of the squares of two natural numbers: 10 = 1 + 9

⇒ 10 = 1² + 3²

On the number line, take OA = 3 units.

Draw BA = 1 units, perpendicular to OA. Join OB.

By Pythagoras theorem, OB = √10 Using a compass with centre O and radius OB, draw an arc which intersects the number line at the point C. Then, C corresponds to √10.

(ii). √17

We can write 17 as the sum of the squares of two natural numbers: 17 = 1 + 16

⇒ 17 = 1² + 4²

On the number line, take OA = 4 units.

Draw BA = 1 units, perpendicular to OA. Join OB.

By Pythagoras theorem, OB = √17 Using a compass with centre O and radius OB, draw an arc which intersects the number line at the point C. Then, C corresponds to √17.