Question

In the figure below, what is the area of the smaller circle?

Statement 1: The two larger circles have same radii of 8 cm each and O'O'' is 12 cm
Statement 2: The centres O, O' and O" of the three circles are collinear

• A. Statement (1) alone is sufficient to answer the question
• B. Statement (2) alone is sufficient to answer the question
• C. Both the statements together are needed to answer the question
• D. Either statement (1) alone or statement (2) alone is sufficient to answer the question
• E. Neither statement (1) nor statement (2) suffices to answer the question.

Answer 1

The Correct Option is C

To determine the area of the smaller circle, we need to utilize the given statements and the figure provided. 

Given:

• The radius of the larger circles is 8 cm each.
• The distance between the centers of the larger circles, \(O'\) and \(O''\), is 12 cm.
• The centers \(O\), \(O'\), and \(O''\) are collinear.

Explanation:

1. Since the centers \(O'\) and \(O''\) have the same radius of 8 cm, and their distance is 12 cm, there is an overlap creating a space for the smaller circle.
2. The smaller circle will have its center \(O\) positioned such that it fits perfectly between the two larger circles.
3. Using the property of collinearity from Statement 2, place center \(O\) between \(O'\) and \(O''\). The total distance between \(O'\) and \(O''\) is partitioned equally by center \(O\).
4. The radius of the smaller circle will be \(\frac{12}{2} = 6\) cm because it must fit snugly between the radii of the two larger circles extending to their centers.
5. Now we can calculate the area of the smaller circle using the formula for the area of a circle, \(A = \pi r^2\), where \(r\) is the radius.

Calculation:

\[ \text{Area of the smaller circle} = \pi \times (6)^2 = 36\pi \text{ cm}^2 \]

Conclusion:

Based on the computation, both statements together are necessary to solve for the area of the smaller circle as each contributes half of the required information:

• Statement 1 provides the radius of the larger circles and their connection.
• Statement 2 ensures the alignment and helps us calculate the smaller circle's size effectively.

Therefore, both statements together are needed to answer the question.