NCERT Exemplar Solutions
Class 10 - Mathematics - CHAPTER 11: Area Related To Circles - Exercise 11.2

Question 10

Step 1: Recall the formula for the area of a sector in terms of its arc length.

For any circle, the area of a sector is given by:

\( A = \tfrac{1}{2} r \times l \)

where:

• \(A\) = area of the sector (measured in square metres, \(\text{m}^2\))
• \(r\) = radius of the circle (measured in metres, \(\text{m}\))
• \(l\) = length of the arc (measured in metres, \(\text{m}\))

Step 2: Write the condition given in the problem.

The areas of the two sectors are equal. That means:

\( A_1 = A_2 \)

So,

\( \tfrac{1}{2} r_1 l_1 = \tfrac{1}{2} r_2 l_2 \)

which simplifies to:

\( r_1 l_1 = r_2 l_2 \)

Step 3: Check if this condition forces \(l_1 = l_2\).

Notice that the equality only says the product of radius and arc length is the same for both sectors. This does not mean the arc lengths must be equal.

Step 4: Give an example to make it clear.

Suppose:

• Sector 1: radius \(r_1 = 2\,\text{m}\), arc length \(l_1 = 6\,\text{m}\)
• Sector 2: radius \(r_2 = 3\,\text{m}\), arc length \(l_2 = 4\,\text{m}\)

Now, \( r_1 l_1 = 2 \times 6 = 12 \) and \( r_2 l_2 = 3 \times 4 = 12 \).

So the areas of the sectors are equal. But \(l_1 = 6\,\text{m}\) and \(l_2 = 4\,\text{m}\), which are not equal.

Final Step: Therefore, it is not necessary that the arc lengths are equal if the areas are equal.

Answer: False.