the triangle below is isosceles. find the length of side x in simplest radical form with a rational…

Explanation:

Step1: Identify the type of isosceles triangle

It's a right - isosceles triangle. In a right - isosceles triangle, the two legs are of equal length. Let the length of each leg be (a = 3), and we want to find the hypotenuse (x).

Step2: Apply the Pythagorean theorem

The Pythagorean theorem for a right - triangle is (a^{2}+b^{2}=c^{2}). In a right - isosceles triangle (a = b). So, (x^{2}=3^{2}+3^{2}). [x^{2}=9 + 9=18]

Step3: Solve for (x)

Take the square root of both sides: (x=\sqrt{18}). Simplify (\sqrt{18}=\sqrt{9\times2}=3\sqrt{2}). Since the denominator is already rational (in this case, we can consider the denominator as 1), the simplest radical form with a rational denominator is (3\sqrt{2}).

Answer:

(3\sqrt{2})