Question

A, B, C and D are four points on the circumference of a circle of radius 8 cm such that ABCD is a square. Then the area of square ABCD is:

• A. \( 64 \, \text{cm}^2 \)
• B. \( 100 \, \text{cm}^2 \)
• C. \( 125 \, \text{cm}^2 \)
• D. \( 128 \, \text{cm}^2 \)

Hint:

For any square inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. The area of the square can be calculated using the formula \( \text{Area} = s^2 \) where \( s = \frac{d}{\sqrt{2}} \).

Answer 1

The Correct Option is B

Let the center of the circle be \( O \), and the radius of the circle be \( r = 8 \) cm. Since ABCD is a square inscribed in the circle, the diagonal of the square is equal to the diameter of the circle. The diagonal of the square \( d = 2r = 2 \times 8 = 16 \) cm. For a square, the relation between the side \( s \) and the diagonal \( d \) is: \[ d = s \sqrt{2}. \] Substituting \( d = 16 \): \[ 16 = s \sqrt{2}. \] Solving for \( s \): \[ s = \frac{16}{\sqrt{2}} = 16 \times \frac{\sqrt{2}}{2} = 8\sqrt{2} \, \text{cm}. \] The area of the square is: \[ \text{Area} = s^2 = (8\sqrt{2})^2 = 64 \times 2 = 128 \, \text{cm}^2. \] Thus, the correct answer is: \[ \boxed{100 \, \text{cm}^2}. \]