M24-05

Official Solution:


A rectangle is inscribed in a circle of diameter 2. Which of the following can be the area of the rectangle?

I. 0.01

II. 2.00

III. 3.20


A. I only
B. II only
C. III only
D. I and II only
E. II and III only


Observe the diagram provided below:



If the width of the blue rectangle is sufficiently small, its area could indeed be 0.01.

In general, the area of the inscribed rectangle is greater than 0 and less than or equal to the area of the inscribed square, with the inscribed square possessing the greatest area among all rectangles that can be inscribed in a given circle.

Now, since the area of the inscribed square in a circle with a diameter of 2 is 2, then the area of the inscribed rectangle is \(0 \lt area \le 2\). Thus, I and II are possible values for the area. (Otherwise, one can notice that the area of the circle is \(\pi r^2 = \pi \approx 3.14\) and the area of the inscribed rectangle cannot be greater than that, so III is not possible).


Answer: D