Question

1. a square is inscribed in a circle with radius r. what is the area of the square?

a. ( r^2 cdot sqrt{3} )
b. ( 2r^3 )
c. ( 2r^2 )
d. ( r^2 )

Explanation:

Step1: Find the diagonal of the square

When a square is inscribed in a circle, the diagonal of the square is equal to the diameter of the circle. The radius of the circle is \( r \), so the diameter is \( 2r \). Thus, the diagonal of the square \( d = 2r \).

Step2: Relate diagonal to side length of square

For a square with side length \( s \) and diagonal \( d \), we use the Pythagorean theorem: \( d^{2}=s^{2} + s^{2}=2s^{2}\). Substituting \( d = 2r \), we get \( (2r)^{2}=2s^{2} \).

Step3: Solve for side length squared

Simplify \( (2r)^{2}=2s^{2} \): \( 4r^{2}=2s^{2} \). Divide both sides by 2: \( s^{2}=2r^{2} \).

Step4: Find the area of the square

The area of a square \( A = s^{2} \). From step 3, \( s^{2}=2r^{2} \), so the area of the square is \( 2r^{2} \).

Answer:

c. \( 2r^{2} \)