We have seen in Class 7 that any odd number can be written as the difference of two perfect squares. (The lesson, Identities) Usingthis, draw lines of lengths centimetres.

→ Any odd number can be written in the form 2k – 1,k being a natural number.

Now,2k – 1 = k² – (k – 1)²

● For,

2k – 1 = 7

⇒2k = 8

⇒ k = 4

So, √7 = 4² – 3²

So, if we draw a line of 4 units, say CD, then,

Then,

Draw a random line and cut a length of 3c.m. s off it, to have AB.

→ Then draw a normal BA’ on AB.

→ Draw a circular arc with A as the centre and CD as centre, which eventually intersects BA’ at E.

→ Join B and E; and, A and E.

See, In Δ ABE, –

AB⊥ BE

BE² = AE² – AB² = 4² – 3² = 7

⇔ E = √7 c.m.

● And,

2k – 1 = 11

⇒ 2k = 12

⇒ k = 6

So, √11 = √( 6² – 5²)

Draw a random line and cut a length of 5 c.m. s off it, to have AB.

→ Then draw a normal BA’ on AB.

→ Draw a circular arc with A as the centre and CD as centre, which eventually intersects BA’ at E.

→ Join B and E; and,A and E.

See, In Δ ABE, –

AB⊥ BE

BE² = AE² – AB² = 6² – 5² = 11

óBE = √11 c.m.