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
Directions: This question has a problem and two statements numbered (1) and (2) giving certain information. You have to decide if the information given in the statements is sufficient for answering the problem. Indicate your answer

• 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 analyze the given statements and their sufficiency. 

1. From Statement 1, we know:
   • The radii of the two larger circles are 8 cm each.
   • The distance between the centers of the larger circles, \(O'O"\), is 12 cm.
2. From Statement 2, we know:
   • The centers \(O\), \(O'\), and \(O"\) are collinear.
3. Considering both statements together:
   • We understand that the centers are collinear, and the distance \(O'O"\) is 12 cm.
   • To maintain the correct geometry, the smaller circle's center \(O'\) should be equidistant from both larger circle centers, forming a semi-circle from each larger circle to the smaller circle. The distance from the center \(O'\) to either \(O\) or \(O"\) would be needed along with a geometric configuration or additional data to calculate the radius of the smaller circle precisely.

While initially, it might seem enough, both statements only provide a geometric setup, lacking concrete measurement facilities for the small circle's radius.

Thus, the correct answer is: both the statements together are needed to answer the question.