Question

vivian draws a square inscribed in a circle. the circle has a diameter of 2x units. create an expression for the length, in units, of one side of the square.

Explanation:

Step1: Identify the diagonal of the square

A square inscribed in a circle has its diagonal equal to the diameter of the circle. The circle's diameter is \(2x\), so the diagonal of the square \(d = 2x\).

Step2: Relate diagonal to side length of square

For a square with side length \(s\), the diagonal \(d\) and side length \(s\) are related by the Pythagorean theorem: \(d^{2}=s^{2}+s^{2}=2s^{2}\). Substitute \(d = 2x\) into the equation: \((2x)^{2}=2s^{2}\).

Step3: Solve for \(s\)

First, expand \((2x)^{2}\) to get \(4x^{2}=2s^{2}\). Then divide both sides by 2: \(2x^{2}=s^{2}\). Take the square root of both sides: \(s=\sqrt{2x^{2}}\). Simplify the square root: \(s = x\sqrt{2}\) (since \(x\) represents a length, we take the positive root).

Answer:

\(x\sqrt{2}\)