Question

in the figure shown below, a line segment x connects the center of a square to a vertex. if the distance x is 2 feet long, then what is the measure, in feet, of the perimeter of the square?
4
8
8√2
16
16√2
note: figure not drawn to scale

Explanation:

Step1: Recall diagonal - side relationship

The line segment \(x\) from the center of the square to a vertex is half of the square's diagonal. So the diagonal \(d\) of the square is \(d = 2x\). Given \(x = 2\) feet, then \(d=4\) feet.

Step2: Use Pythagorean theorem for square

For a square of side length \(s\), by the Pythagorean theorem \(d^{2}=s^{2}+s^{2}=2s^{2}\). Since \(d = 4\), we have \(4^{2}=2s^{2}\), which simplifies to \(16 = 2s^{2}\), then \(s^{2}=8\), and \(s = 2\sqrt{2}\) feet.

Step3: Calculate the perimeter

The perimeter \(P\) of a square is \(P = 4s\). Substituting \(s = 2\sqrt{2}\) into the formula, we get \(P=4\times2\sqrt{2}=8\sqrt{2}\) feet.

Answer:

C. \(8\sqrt{2}\)