Question

the figure below is a square. find the length of side x in simplest radical form with a rational denominator. (figure of a square with a diagonal x and a side length of 3) answer attempt 1 out of 2 x = blank submit answer

Explanation:

Step1: Identify triangle type

The square is split into two right - isosceles triangles. Each side of the square is 3, and the diagonal \(x\) forms the hypotenuse of a right - triangle with legs of length 3.

Step2: Apply Pythagorean theorem

For a right - triangle with legs \(a = 3\) and \(b = 3\) and hypotenuse \(x\), the Pythagorean theorem states that \(x^{2}=a^{2}+b^{2}\). Substituting \(a = 3\) and \(b = 3\) into the formula, we get \(x^{2}=3^{2}+3^{2}=9 + 9=18\).

Step3: Simplify the radical

To find \(x\), we take the square root of both sides: \(x=\sqrt{18}\). We can simplify \(\sqrt{18}\) as \(\sqrt{9\times2}=\sqrt{9}\times\sqrt{2}=3\sqrt{2}\) (since the denominator is already rational, and we have the simplest radical form).

Answer:

\(3\sqrt{2}\)