Question 31 of 58intermediate💡 UnderstandShort Answer2 marks

We have seen how to obtain a line whose length is a rational number. How do we obtain lines whose lengths are irrational?

Correct Answer

To obtain lines whose lengths are irrational, such as 2\sqrt{2}, we can use the Pythagorean theorem and a compass. First, on the number line, measure OA = 1 unit and draw a perpendicular at A.

Next, on this perpendicular, mark a point B such that AB = 1 unit. Join the origin O to B. The length of OB will be 12+12=2\sqrt{1^2 + 1^2} = \sqrt{2} units, which is an irrational length.

Finally, using a compass with O as the centre and OB as the radius, draw an arc that intersects the number line at P. The length OP equals 2\sqrt{2} units, successfully representing the irrational number on the number line. This method can be generalised for any n\sqrt{n}.

Exercise: Think and Reflect | Q: 1 | (Page 15)
For More Understanding

Explanation

The textbook explains the construction of irrational lengths like 2\sqrt{2} using right triangles. By creating a right triangle with legs of 1 unit each, the hypotenuse becomes 2\sqrt{2}. This hypotenuse is then used as the radius of an arc drawn from the origin to intersect the number line, thereby locating the irrational number. This approach can be extended to construct 3\sqrt{3}, 5\sqrt{5}, and other lengths of the form √n.

Solution Steps

  1. Step 1: On the number line, measure OA = 1 unit and draw a perpendicular at A.

  2. Step 2: Mark point B on the perpendicular such that AB = 1 unit. Join O to B. OB = 2\sqrt{2} units.

  3. Step 3: With O as the centre and OB as the radius, draw an arc intersecting the number line at P. OP = 2\sqrt{2} units.

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