Question

Represent `sqrt(10)` on the number line.

Answer 1

Given: Represent `sqrt(10)` on the number line.

Stepwise calculation:

1. Identify two perfect squares between which 10 lies 9 < 10 < 16.

So, `sqrt(9) < sqrt(10) < sqrt(16)`.

That is, `3 < sqrt(10) < 4`.

2. On the number line, mark points 3 and 4.

3. To locate `sqrt(10)` approximately, find a point between 3 and 4 whose square is close to 10. 

For example,

3.1² = 9.61

3.2² = 10.24

So, `sqrt(10)` lies between 3.1 and 3.2.

4. Use geometric construction:

Draw a segment of length 3 units on the number line representing point 3.

At point 4, draw a line segment perpendicular to the number line of length 1 unit because 4 – 3 = 1.

Connect the endpoint of this perpendicular line to point 3.

According to the Pythagoras theorem, the length of the hypotenuse of this right triangle is `sqrt(3^2 + 1^2) = sqrt(10)`.

With the compass, mark this length from 0 along the number line starting at 0. This point is `sqrt(10)`.

`sqrt(10)` is represented on the number line as the point between 3 and 4 such that the distance from 0 to this point is the hypotenuse of a right triangle with legs 3 and 1 units, i.e., of length `sqrt(10)`.