Show how √3 can be represented on the number line.

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Detailed Solution

Let line AB be of 1 unit on a number line.
At B, draw a perpendicular line BC of length 1 unit.
Join CA.
Now, ABC is a right-angled triangle. Applying Pythagoras' theorem:
- AB² + BC² = CA²
- 1² + 1² = CA²
- CA² = 2
- ⇒ CA = √2
Thus, CA is a line of length √2 units.
Taking CA as the base, draw a perpendicular line from point A of length 1 unit. Label the endpoint of this line as D.
Join CD.
Now, ADC is a right-angled triangle. Applying Pythagoras' theorem again:
- CA² + AD² = CD²
- (√2)² + 1² = CD²
- 2 + 1 = CD²
- CD² = 3
- ⇒ CD = √3
Thus, CD is a line of length √3 units.
Taking CD as a radius and D as the center, draw an arc touching the number line. The point at which the number line is intersected by the arc is at √3 distance from 0 because it is the radius of the circle whose center is D.

Thus, √3 is represented on the number line as shown in the figure.

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