Bunuel wrote:
The above picture depicts a circle perfectly inscribed in a square. Is the shaded region > 4?
(1) The area of the large rectangle equals 64.
(2) The perimeter of the shaded region equals 8 + 2π.
Kudos for a correct solution.Attachment:
circle-in-square-red2.GIF
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.