If a rectangle with length sqrt(a) units and width sqrt(b)

Diagonal of rectangle is equal to the diameter of circle.

Here is the thing though, why can't you have rectangles of different side dimensions inscribed in the circle as in my attached image?

Just for kicks, I drew the attached image out with a pen, paper, a circular template and a ruler. I understand that the radius is 5 (so diameter is 10) but if we drew a rectangle with almost no height it would be just slightly longer while being a lot more narrow. The ratio of length to width wouldn't change enough to make them the same area.

EDIT: Thinking about it, it makes a bit more sense. They're testing to see whether you completely understand the concept of a right triangle. In a right triangle (which this is, it's two of them) a^2 + b^2 MUST EQUAL c^2. Drawing it out throws you off. If you inscribe a rectangle in a circle and you change the dimensions of that rectangle while still keeping it inscribed, the change in width will offset the change in height and vice versa. I guess my hand drawn attempt at understanding this was incorrect. In other words:

a^2 + b^2 = c^2
6^2 + 8^2 = 100
36 + 64 = 100

(a-b)^2 + (b+a)^2 = 100 as well.

Simple, yet very tricky.

Bunuel

Is it okay to assume that when we create a hypotenuse for a rectangle we cut it into a 30-60-90 triangle?

Hence \(\sqrt{a} = 5\) and \(\sqrt{b} = 5\sqrt{3}\)?

The diagonal of the rectangle is 2*R = 2*5=10

Applying the Pythagorean Theorem gives=

10^2 = (\(\sqrt{a}\))^2 + (\(\sqrt{b}\))^2

100= a+b

E.