A rectangle is inscribed in a circle of diameter 2. Which of

If the length of a diagonal of a square is 2, then \(side^2+side^2= diagonal^2\) --> \(2*side^2=2^2\) --> \(side^2=area=2\).

You could get the side in another way: since the angle between a diagonal and a side in a square is 45 degrees, then \(side=\frac{diagonal}{\sqrt{2}}\) (from the properties of 45-45-90 triangle).

You could also get the area directly: \(area_{square}=\frac{diagonal^2}{2}=2\).

Complete 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

Look at the diagram below:
If the width of blue rectangle is small enough then its area could be 0.01.

Generally, the are of the inscribed rectangle is more than 0 and less than or equal to the area of the inscribed square (inscribed square has the largest area from all rectangles that can be inscribed in a given circle).

Now, since the area of the inscribed square in a circle with the diameter of 2 is 2, then the area of the inscribed rectangle is \(0<area\leq{2}\). So, I and II are possible values of the area. (Else you 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.