Question

1. in the figure, the circle circumscribed about the square abcd has a circumference of 8π. find the area of the square abcd. a 4√2 b 32 c 32π d it cannot be determined from the information given.

Explanation:

Step1: Find the diameter of the circle

The circumference of a circle is given by \( C = \pi d \), where \( d \) is the diameter. We know the circumference \( C = 8\pi \). So, \( 8\pi=\pi d \). Dividing both sides by \( \pi \), we get \( d = 8 \).

Step2: Recognize the diameter as the diagonal of the square

Since the circle is circumscribed about the square, the diagonal of the square \( ABCD \) is equal to the diameter of the circle, so the diagonal \( d_{square}=8 \).

Step3: Find the side length of the square

For a square with diagonal \( d \), the side length \( s \) is related to the diagonal by the formula \( d = s\sqrt{2} \). So, \( s=\frac{d}{\sqrt{2}} \). Substituting \( d = 8 \), we get \( s=\frac{8}{\sqrt{2}} = 4\sqrt{2} \) (after rationalizing the denominator: \( \frac{8}{\sqrt{2}}\times\frac{\sqrt{2}}{\sqrt{2}}=\frac{8\sqrt{2}}{2}=4\sqrt{2} \)).

Step4: Calculate the area of the square

The area of a square is \( A = s^{2} \). Substituting \( s = 4\sqrt{2} \), we get \( A=(4\sqrt{2})^{2}=16\times2 = 32 \).

Answer:

32