Lab Manual: Construct a Square Root Spiral | Lab Manuals for Class 9 PDF Download

Objective

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.