Final answer:
The largest rectangle that can be inscribed in a circle of radius r is a square with side lengths of √2 × r.
Step-by-step explanation:
The problem at hand is to determine the dimensions of the largest rectangle that can be inscribed in a circle of a given radius r.
The largest area for such a rectangle within a circle occurs when the rectangle is actually a square, as the square has all sides of equal length and can be inscribed in the circle with its diagonal equal to the diameter of the circle. Thus, for a circle of radius r, the largest inscribed rectangle is a square with a side length of √2 × r.
To arrive at this answer, we consider that the diagonal of the square must be equal to the diameter of the circle (2r), and since the diagonal of the square is the hypotenuse of a right-angled triangle whose sides are equal (as the square's sides are equal), we can use the Pythagorean theorem to find the length of the side of the square.
If one side of the square is represented as x, then we have 2x^2 = (2r)^2, leading us to x = r√2. Therefore, in non-decreasing order, the dimensions of the rectangle are √2 × r and √2 × r.