NCERT Class 9 Maths Lab Manual – Construct a Square Root Spiral
To construct a square root spiral.
- Geometry box
- A piece of plywood
- Concept of number line.
- Concept of irrational numbers.
- Pythagoras theorem.
- 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
- 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’.
- 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
=> PG = √6 and so on.
On actual measurement, we get
PC = …….. ,
PD = …….. ,
PE = …….. ,
PF = …….. ,
PG = …….. ,
√2 = PC = …. (approx.)
√3 = PD = …. (approx.)
√4 = PE = …. (approx.)
√5 = PF = …. (approx.)
A square root spiral has been constructed.
With the help of explained activity, existence of irrational numbers can be illustrated.
Define a rational number.
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.
Define an irrational number.
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.
Define a real number.
A number which may be either rational or irrational is called a real number.
How many rational and irrational numbers lie between any two real numbers?
There are infinite rational and irrational numbers lie between any two real numbers.
Is it possible to represent irrational numbers on the number line?
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.
In which triangle, Pythagoras theorem is applicable?
Right angled triangle
Give some examples of irrational numbers.
Some examples of irrational numbers are √5, 3 – √7,2π, etc.
Can we represent the reciprocal of zero on the number line.
No, because reciprocal of zero is undefined term, so we cannot represent on number line.
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)?
Is it possible that we make a square root spiral of negative nymbers?
Represent square root of 7 and 9 by constructing a square root spiral.