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. If the data in statement (1) alone is sufficient to answer the question
• B. If the data in statement (2) alone is sufficient to answer the question
• C. If the data in both the statements together are needed to answer the question
• D. If either statement (1) alone or statement (2) alone is sufficient to answer the question

Hint:

Think about what the small circle's diameter equals along the line joining the centres.

Answer 1

The Correct Option is C

Step 1: Understand the figure.
Three circles sit on one line. Two larger circles have centres O' and O''.
The smaller circle sits in the overlap and has centre O.

Step 2: Check statement 1 alone.
Statement 1 gives the radii of the two larger circles as 8 cm each and the distance O'O'' as 12 cm.
It does not say the centre O of the small circle lies on the line O'O''.
Without that fact we cannot say the small circle's diameter equals the overlap length measured along that line.
So statement 1 alone is not sufficient.

Step 3: Check statement 2 alone.
Statement 2 only says the three centres O, O' and O'' are collinear.
It gives no radius or distance value, so no area can be found from this fact alone.
So statement 2 alone is not sufficient.

Step 4: Combine both statements.
With both larger circles having radius 8 cm and O'O'' = 12 cm, the overlap along the common line has length \( 8 + 8 - 12 = 4 \) cm.
Since statement 2 confirms O also lies on this same line, the small circle's diameter equals this overlap length, 4 cm.
So the small circle's radius is 2 cm and its area is \( \pi (2)^2 = 4\pi \) sq cm.

Final Answer:
Both statements together are needed to find the area of the smaller circle. \[ \boxed{\text{Both statements together are needed (option c)}} \]