NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 10: Construction - Exercise 10.2

Question 1

Step 1: We are asked to divide a line segment in the ratio \(\sqrt{3} : \dfrac{1}{\sqrt{3}}\).

Step 2: To make the ratio easier to understand, we remove the square root in the denominator. Multiply both sides of the ratio by \(\sqrt{3}\):

\[ \sqrt{3} : \dfrac{1}{\sqrt{3}} = (\sqrt{3} \times \sqrt{3}) : \left(\dfrac{1}{\sqrt{3}} \times \sqrt{3}\right) = 3 : 1 \]

Step 3: Now the ratio is written as \(3:1\). This is a ratio using whole numbers (integers).

Step 4: In geometry, to divide a line segment in the ratio \(m:n\), where \(m\) and \(n\) are whole numbers, we can use the standard geometrical construction method:

• Draw a ray at one end of the line segment.
• Mark \(m+n\) equal parts on that ray (here, \(3+1 = 4\) equal parts).
• Join the last point of the ray to the other end of the line segment.
• Draw a line parallel through the 3rd mark (since the ratio is 3:1).
• This point of intersection divides the line segment in the ratio \(3:1\).

Final Step: Since \(\sqrt{3} : \dfrac{1}{\sqrt{3}}\) becomes \(3:1\), it is possible to divide the line segment using this construction.