The above picture depicts a circle perfectly inscribed in a square. Is

Hi Bunuel:
Can you please advise? How come in the same question we get different answers.
In statement 1: r=4\sqrt{2}
In Statement 2: r =4
Clearly, the question is flawed or do I miss something?

GROCKIT OFFICIAL SOLUTION:

Statement 1: By knowing the area of the large square we also know the lengths of its sides. (Note that 64 is a perfect square, which should be a clue.) If the side is 8, then so is the diameter, which means the radius equals 4. In the image, we can see that the “larger figure” is the top-right square bordered by two radii and the outer border. How do we know that it’s a square? Two pieces of information: All sides are equal to 4, and the radius meets the large square at a right angle because it is a tangent. In this instance, the area of the smaller square equals 16. Since the interior angle is 90-degrees (360/4), the area of the sector of the circle can be represented by A = πr²/4. So that A = 16π/4 = 4π.

A(shaded) = A(small square) – A(sector) = 16 – 4π < 4 because 4π > 12. Sufficient.

Statement 2: The first thing that should jump out is the combination of a π-term and non- π-term. We can reasonably assume that 8 represents the two straight sides of the perimeter and the 2π the arc-length of the quarter circle, which is known because of the internal right angle. If 2π = C/4, then C = 8π. If C = 8π, then r = 4. From here, we return to the same reasoning as above:

A(shaded) = A(small square) – A(sector) = 16 – 4π < 4 because 4π > 12. Sufficient.

Each statement is sufficient, so the answer is choice D.

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