**NCERT Class 9 Maths Lab Manual – Construct a Square Root Spiral**

**OBJECTIVE**

To construct a square root spiral.

**Materials Required**

- Adhesive
- Geometry box
- Marker
- A piece of plywood

**Prerequisite Knowledge**

- Concept of number line.
- Concept of irrational numbers.
- Pythagoras theorem.

**Theory**

- A number line is a imaginary line whose each point represents a real number.
- The numbers which cannot be expressed in the form p/q where q ≠ 0 and both p and q are integers, are called irrational numbers, e.g. √3, π, etc.
- According to Pythagoras theorem, in a right angled triangle, the square of the hypotenuse is equal to the sum of the squares of other two sides containing right angle. ΔABC is a right angled triangle having right angle at B. (see Fig. 1.1)

Therefore, AC² = AB² +BC²

where, AC = hypotenuse, AB = perpendicular and BC = base

**Procedure**

- Take a piece of plywood having the dimensions 30 cm x 30 cm.
- Draw a line segment PQ of length 1 unit by taking 2 cm as 1 unit, (see Fig. 1.2)

- Construct a line QX perpendicular to the line segment PQ, by using compasses or a set square, (see Fig. 1.3)

- From Q, draw an arc of 1 unit, which cut QX at C(say). (see Fig. 1.4)

- Join PC.
- Taking PC as base, draw a perpendicular CY to PC, by using compasses or a set square.
- From C, draw an arc of 1 unit, which cut CY at D (say).
- Join PD. (see Fig. 1.5)

- Taking PD as base, draw a perpendicular DZ to PD, by using compasses or a set square.
- From D, draw an arc of 1 unit, which cut DZ at E (say).
- Join PE. (see Fig. 1.5)

Keep repeating the above process for sufficient number of times. Then, the figure so obtained is called a ‘square root spiral’.

**Demonstration**

- In the Fig. 1.5, ΔPQC is a right angled triangle.

So, from Pythagoras theorem,

we have PC² = PQ² + QC²

[∴ (Hypotenuse)² = (Perpendicular)² + (Base)²]

= 1² +1² =2

=> PC = √2

Again, ΔPCD is also a right angled triangle.

So, from Pythagoras theorem,

PD² =PC² +CD²

= (√2)² +(1)² =2+1 = 3

=> PD = √3 - Similarly, we will have

PE= √4

=> PF=√5

=> PG = √6 and so on.

**Observations**

On actual measurement, we get

PC = …….. ,

PD = …….. ,

PE = …….. ,

PF = …….. ,

PG = …….. ,

√2 = PC = …. (approx.)

√3 = PD = …. (approx.)

√4 = PE = …. (approx.)

√5 = PF = …. (approx.)

**Result**

A square root spiral has been constructed.

**Application**

With the help of explained activity, existence of irrational numbers can be illustrated.

**Viva Voce**

**Question 1:**

Define a rational number.

**Answer:**

A number which can be expressed in the form of p/q, where q ≠ 0 and p, q are integers, is called a rational number.

**Question 2:**

Define an irrational number.

**Answer:**

A number which cannot be expressed in the form of p/q, where q ≠ 0 and p, q are integers, is called an irrational number.

**Question 3:**

Define a real number.

**Answer:**

A number which may be either rational or irrational is called a real number.

**Question 4:**

How many rational and irrational numbers lie between any two real numbers?

**Answer:**

There are infinite rational and irrational numbers lie between any two real numbers.

**Question 5:**

Is it possible to represent irrational numbers on the number line?

**Answer:**

Yes, as we know that each point on the number line represent a real number (i.e. both rational and irrational), so irrational number can be represented on number line.

**Question 6:**

In which triangle, Pythagoras theorem is applicable?

**Answer:**

Right angled triangle

**Question 7:**

Give some examples of irrational numbers.

**Answer:**

Some examples of irrational numbers are √5, 3 – √7,2π, etc.

**Question 8:**

Can we represent the reciprocal of zero on the number line.

**Answer:**

No, because reciprocal of zero is undefined term, so we cannot represent on number line.

**Question 9:**

In a square root spiral, is it true that in each square root of natural number is equal to the square root of the sum of 1 and previous natural number (> 1)?

**Answer:**

Yes

**Question 10:**

Is it possible that we make a square root spiral of negative nymbers?

**Answer:**

No

**Suggested Activity**

Represent square root of 7 and 9 by constructing a square root spiral.

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