Question

use the figure below to answer question 16.

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

Since \(ABCD\) is a square inscribed in a circle and \(\angle BAD = 90^{\circ}\), \(BD\) is the diameter of the circle. Given the circumference of the circle \(C=\pi d = 8\pi\), we can solve for \(d\) (diameter) using the formula \(C=\pi d\). So, \(d=\frac{C}{\pi}=\frac{8\pi}{\pi}=8\).

Step2: Find the side length of the square

Let the side length of the square \(ABCD\) be \(a\). Using the Pythagorean theorem in right - triangle \(ABD\) (\(BD\) is the hypotenuse), \(BD^{2}=AB^{2}+AD^{2}\). Since \(AB = AD=a\) (sides of a square) and \(BD = 8\), we have \(8^{2}=a^{2}+a^{2}\), \(64 = 2a^{2}\), \(a^{2}=32\).

Step3: Calculate the area of the square

The area of a square \(A=a^{2}\). Since \(a^{2}=32\), the area of square \(ABCD\) is \(32\).

Answer:

B. \(32\)