1. Find the area of a square with a diagonal of 8 inches.

To find the area of a square given its diagonal, we can use the relationship between the diagonal \( d \) and the side length \( s \) of the square. The formula for the diagonal of a square is: \[ d = s\sqrt{2} \] From this, we can express the side length \( s \) in terms of the diagonal \( d \): \[ s = \frac{d}{\sqrt{2}} \] Once we have the side length, we can find the area \( A \) of the square using the formula: \[ A = s^2 \] Now, let's calculate the side length and then the area step by step. 1. Given \( d = 8 \) inches, we can find \( s \): \[ s = \frac{8}{\sqrt{2}} \] 2. Now, we can calculate \( s^2 \) to find the area \( A \): \[ A = s^2 = \left(\frac{8}{\sqrt{2}}\right)^2 \] Let's perform the calculations. First, calculate \( s \): \[ s = \frac{8}{\sqrt{2}} = 8 \cdot \frac{\sqrt{2}}{2} = 4\sqrt{2} \] Now, calculate the area \( A \): \[ A = (4\sqrt{2})^2 = 16 \cdot 2 = 32 \] Thus, the area of the square is \( 32 \) square inches.