A gardener wants to divide a square piece of lawn in half diagonally. What is the length of the diagonal if the side of the square is 8 ft? Leave your answer in simplest radical form.
The diagonal of a square can be calculated using the Pythagorean theorem, which states that in a right triangle, the square of the length of the hypotenuse (diagonal) is equal to the sum of the squares of the other two sides.
In this case, the two sides of the square are both 8 ft, so the diagonal can be calculated as follows:
\( \text{Diagonal}^2 = 8^2 + 8^2 \)
\( \text{Diagonal}^2 = 64 + 64 \)
\( \text{Diagonal}^2 = 128 \)
Taking the square root of both sides to find the diagonal:
\( \text{Diagonal} = \sqrt{128} \)
\( \text{Diagonal} = \sqrt{64 \times 2} \)
\( \text{Diagonal} = \sqrt{64} \times \sqrt{2} \)
\( \text{Diagonal} = 8\sqrt{2} \)
Therefore, the length of the diagonal when the side of the square is 8 ft is \(8\sqrt{2}\) ft.