Question

If rectangles are inscribed in a circle of radius \( r \) units, then the dimensions of the rectangle which has maximum area are

• A. \( 2r \) units, \( r \) units
• B. \( 2r \) units, \( \sqrt{2}r \) units
• C. \( r \) units, \( \sqrt{2}r \) units
• D. \( \sqrt{2}r \) units, \( \sqrt{2}r \) units

Hint:

For a rectangle inscribed in a circle, the maximum area occurs when the rectangle is a square, and its side length is \( \sqrt{2}r \).

Answer 1

The Correct Option is D

Step 1: Understand the problem.
The area of a rectangle inscribed in a circle is maximized when the rectangle is a square. The diagonal of the square is the diameter of the circle.
Step 2: Use the relationship between the diagonal and the sides of the square.
For a square inscribed in a circle, the diagonal is equal to the diameter of the circle, which is \( 2r \). The sides of the square, \( s \), satisfy: \[ s \sqrt{2} = 2r \] Thus: \[ s = \frac{2r}{\sqrt{2}} = \sqrt{2}r \]
Step 3: Conclusion.
The dimensions of the rectangle with maximum area are \( \sqrt{2}r \) units by \( \sqrt{2}r \) units.