Question:medium

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.

Show Hint

Think about what the small circle's diameter equals along the line joining the centres.
Updated On: Jul 21, 2026
  • If the data in statement (1) alone is sufficient to answer the question
  • If the data in statement (2) alone is sufficient to answer the question
  • If the data in both the statements together are needed to answer the question
  • If either statement (1) alone or statement (2) alone is sufficient to answer the question
Show Solution

The Correct Option is C

Solution and Explanation

Step 1: Set up a number line for the centres.
Place O' and O'' on a straight line, since the figure shows all three centres in a row.

Step 2: Test statement 1 by itself.
Statement 1 fixes both larger radii at 8 cm and the gap O'O'' at 12 cm.
But this alone does not confirm the small circle's centre O sits exactly on the O'O'' line.
Without that placement, the 12 cm gap cannot be turned into the small circle's size.
Statement 1 by itself fails to fix the area.

Step 3: Test statement 2 by itself.
Statement 2 only fixes that O, O' and O'' fall on one straight line.
There is no length or radius attached to this fact, so no numeric area follows.
Statement 2 by itself also fails.

Step 4: Merge the two statements.
Put O' at 0 and O'' at 12 on this line, with each larger circle carrying radius 8.
The left circle then spans from -8 to 8, and the right circle spans from 4 to 20.
Their shared stretch on the line runs from 4 to 8, a length of 4 cm.
Because O lies on this same line, that 4 cm stretch is exactly the small circle's diameter.
This gives radius 2 cm and area $ \pi \times 2^2 = 4\pi $ sq cm.

Final Answer:
Only the merged data from both statements pins down the area. \[ \boxed{\text{Option (c): both statements together}} \]
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