State if the statement is true or false: The representation of

2

,

3

and

5

on the real number line is

Question

State if the statement is true or false: The representation of

2

,

3

and

5

on the real number line is

Accepted Answer

Let us represent

2

first, on the number line. To do so, mark a point 1 unit away from the starting point and then draw a perpendicular on that line, of 1 unit again. Next, join the initial point with the free end of the perpendicular to form a triangle as shown below,Applying Pythagoras theorem on the above triangle, we get the length of the hypotenuse as follows,

a
      2
     
     +
      b
      2
     
     =
      c
      2

1
      2
     
     +
      1
      2
     
     =
      c
      2

c
      2
     
     =2

c=
      2

Now, taking the hypotenuse as the radius and the initial point as the center, draw an arc that intersects the number line. The point where it intersects is

2

, as shown above in the diagram.Next, to represent

3

on the number line, repeat the same procedure as for

2

, i.e., on the same triangle, considering the hypotenuse as the base, draw a perpendicular line of 1 unit and join it to the initial point as shown below. Applying the Pythagoras theorem on the second triangle, we get,

a
      2
     
     +
      b
      2
     
     =
      c
      2

(
       2
      
      )
      2
     
     +
      1
      2
     
     =
      c
      2

c
      2
     
     =2+1

c=
      3

Now, taking the second hypotenuse as the radius and the initial point as the center, draw an arc that intersects the number line. The point where it intersects is

3

, as shown above in the diagram.Now, for

5

, take the point two units away from the initial point, i.e., 0, and draw a perpendicular of 1 unit. Then, join the free end of the perpendicular to the initial point as and form a triangle as shown below.Applying the Pythagoras theorem on the above triangle, we get,

a
      2
     
     +
      b
      2
     
     =
      c
      2

2
      2
     
     +
      1
      2
     
     =
      c
      2

c
      2
     
     =4+1

c=
      5

Now, taking the center as the initial point and radius as the hypotenuse, draw an arc that intersects the number line.The point where it intersects is

5

, as shown above.So, the given representation is correct.Hence, the statement is true.So, the correct option is 1.

State if the statement is true or false: The representation of $2$ , $3$ and $5$ on the real number line is

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