Final answer:
The maximum area of a rectangle inside a circle can be found by using a square inscribed in the circle. The area is independent of the circle's radius and the rectangle's maximum area is half of the circle's area. The rectangle with the maximum area touches the circle at its center.
Step-by-step explanation:
To find the maximum area of a rectangle inside a circle, we can use a square inscribed in the circle. The maximum area occurs when the rectangle is a square. This is because a square has equal sides, so its area will be the largest possible.
The area of a rectangle is independent of the circle's radius. The size of the rectangle only depends on the shape of the rectangle itself, not the size of the circle it is inscribed in.
The rectangle's maximum area is half of the circle's area. This is because a square inscribed in a circle divides the circle into four equal parts, and the area of the square is equal to one of those parts.
The rectangle with the maximum area touches the circle at its center. This is because a square inscribed in a circle has its vertices touching the circumference of the circle.